measure_theory.function.locally_integrable ⟷ Mathlib.MeasureTheory.Function.LocallyIntegrable

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -105,7 +105,7 @@ theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
     have : s ⊆ ⋃ x : s, u x := fun y hy => mem_Union_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
     obtain ⟨T, hT_count, hT_un⟩ := is_open_Union_countable u u_open
     refine' ⟨T, hT_count, _⟩
-    rw [← hT_un, bUnion_eq_Union] at this 
+    rw [← hT_un, bUnion_eq_Union] at this
     rw [← Union_inter, eq_comm, inter_eq_right_iff_subset]
     exact this
   have : Countable T := countable_coe_iff.mpr T_count
@@ -196,7 +196,7 @@ theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace
     refine' ⟨u, hu_o.mem_nhds hu_x, _⟩
     rw [integrable_on, restrict_restrict hu_o.measurable_set]
     exact ht_int.mono_set hu_sub
-  · rw [← isOpen_compl_iff] at hs 
+  · rw [← isOpen_compl_iff] at hs
     refine' ⟨sᶜ, hs.mem_nhds h, _⟩
     rw [integrable_on, restrict_restrict, inter_comm, inter_compl_self, ← integrable_on]
     exacts [integrable_on_empty, hs.measurable_set]

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -52,11 +52,11 @@ def LocallyIntegrableOn (f : X → E) (s : Set X)
 #align measure_theory.locally_integrable_on MeasureTheory.LocallyIntegrableOn
 -/
 
-#print MeasureTheory.LocallyIntegrableOn.mono /-
-theorem LocallyIntegrableOn.mono (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
+#print MeasureTheory.LocallyIntegrableOn.mono_set /-
+theorem LocallyIntegrableOn.mono_set (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) : LocallyIntegrableOn f t μ := fun x hx =>
   (hf x <| hst hx).filter_mono (nhdsWithin_mono x hst)
-#align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono
+#align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono_set
 -/
 
 #print MeasureTheory.LocallyIntegrableOn.norm /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,7 +3,7 @@ Copyright (c) 2022 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
-import Mathbin.MeasureTheory.Integral.IntegrableOn
+import MeasureTheory.Integral.IntegrableOn
 
 #align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
 
@@ -129,7 +129,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
       let ⟨K, hK, h2K⟩ := WeaklyLocallyCompactSpace.exists_compact_mem_nhds x
       ⟨_, inter_mem_nhdsWithin s h2K,
         hf _ (inter_subset_left _ _)
-          (isCompact_of_isClosed_subset hK (hs.inter hK.IsClosed) (inter_subset_right _ _))⟩
+          (IsCompact.of_isClosed_subset hK (hs.inter hK.IsClosed) (inter_subset_right _ _))⟩
   · obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
     refine' ⟨K, _, hf K h3K hK⟩
     simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -400,7 +400,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   · exact integrable_on_empty
   have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
   have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
-  have : Metric.Bounded (f '' s) := Metric.bounded_of_bddAbove_of_bddBelow habove hbelow
+  have : Bornology.IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
   rcases bounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
   have A : integrable_on (fun x => C) s μ := by
     simp only [hs.lt_top, integrable_on_const, or_true_iff]

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -126,7 +126,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
   cases hs
   ·
     exact
-      let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
+      let ⟨K, hK, h2K⟩ := WeaklyLocallyCompactSpace.exists_compact_mem_nhds x
       ⟨_, inter_mem_nhdsWithin s h2K,
         hf _ (inter_subset_left _ _)
           (isCompact_of_isClosed_subset hK (hs.inter hK.IsClosed) (inter_subset_right _ _))⟩
@@ -234,7 +234,7 @@ theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f
 theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact k → IntegrableOn f k μ :=
   ⟨fun hf k hk => hf.integrableOn_isCompact hk, fun hf x =>
-    let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
+    let ⟨K, hK, h2K⟩ := WeaklyLocallyCompactSpace.exists_compact_mem_nhds x
     ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
-
-! This file was ported from Lean 3 source module measure_theory.function.locally_integrable
-! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.MeasureTheory.Integral.IntegrableOn
 
+#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
+
 /-!
 # Locally integrable functions
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -55,33 +55,44 @@ def LocallyIntegrableOn (f : X → E) (s : Set X)
 #align measure_theory.locally_integrable_on MeasureTheory.LocallyIntegrableOn
 -/
 
+#print MeasureTheory.LocallyIntegrableOn.mono /-
 theorem LocallyIntegrableOn.mono (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) : LocallyIntegrableOn f t μ := fun x hx =>
   (hf x <| hst hx).filter_mono (nhdsWithin_mono x hst)
 #align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.norm /-
 theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
     LocallyIntegrableOn (fun x => ‖f x‖) s μ := fun t ht =>
   let ⟨U, hU_nhd, hU_int⟩ := hf t ht
   ⟨U, hU_nhd, hU_int.norm⟩
 #align measure_theory.locally_integrable_on.norm MeasureTheory.LocallyIntegrableOn.norm
+-/
 
+#print MeasureTheory.IntegrableOn.locallyIntegrableOn /-
 theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyIntegrableOn f s μ :=
   fun x hx => ⟨s, self_mem_nhdsWithin, hf⟩
 #align measure_theory.integrable_on.locally_integrable_on MeasureTheory.IntegrableOn.locallyIntegrableOn
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact /-
 /-- If a function is locally integrable on a compact set, then it is integrable on that set. -/
 theorem LocallyIntegrableOn.integrableOn_isCompact (hf : LocallyIntegrableOn f s μ)
     (hs : IsCompact s) : IntegrableOn f s μ :=
   IsCompact.induction_on hs integrableOn_empty (fun u v huv hv => hv.mono_set huv)
     (fun u v hu hv => integrableOn_union.mpr ⟨hu, hv⟩) hf
 #align measure_theory.locally_integrable_on.integrable_on_is_compact MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset /-
 theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t μ :=
   (hf.mono hst).integrableOn_isCompact ht
 #align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable /-
 theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) :=
   by
@@ -104,7 +115,9 @@ theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
   rw [hT, aestronglyMeasurable_iUnion_iff]
   exact fun i : T => (hu i).AEStronglyMeasurable
 #align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable
+-/
 
+#print MeasureTheory.locallyIntegrableOn_iff /-
 /-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
 every compact subset contained in `s`. -/
 theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
@@ -124,6 +137,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
     refine' ⟨K, _, hf K h3K hK⟩
     simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
 #align measure_theory.locally_integrable_on_iff MeasureTheory.locallyIntegrableOn_iff
+-/
 
 end LocallyIntegrableOn
 
@@ -137,13 +151,17 @@ def LocallyIntegrable (f : X → E) (μ : Measure X := by exact MeasureTheory.Me
 #align measure_theory.locally_integrable MeasureTheory.LocallyIntegrable
 -/
 
+#print MeasureTheory.locallyIntegrableOn_univ /-
 theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ := by
   simpa only [locally_integrable_on, nhdsWithin_univ, mem_univ, true_imp_iff]
 #align measure_theory.locally_integrable_on_univ MeasureTheory.locallyIntegrableOn_univ
+-/
 
+#print MeasureTheory.LocallyIntegrable.locallyIntegrableOn /-
 theorem LocallyIntegrable.locallyIntegrableOn (hf : LocallyIntegrable f μ) (s : Set X) :
     LocallyIntegrableOn f s μ := fun x hx => (hf x).filter_mono nhdsWithin_le_nhds
 #align measure_theory.locally_integrable.locally_integrable_on MeasureTheory.LocallyIntegrable.locallyIntegrableOn
+-/
 
 #print MeasureTheory.Integrable.locallyIntegrable /-
 theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable f μ := fun x =>
@@ -151,6 +169,7 @@ theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable
 #align measure_theory.integrable.locally_integrable MeasureTheory.Integrable.locallyIntegrable
 -/
 
+#print MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict /-
 /-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
 (See `locally_integrable_on_iff_locally_integrable_restrict` for an iff statement when `s` is
 closed.) -/
@@ -164,7 +183,9 @@ theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace
   simpa only [integrable_on, measure.restrict_restrict hu_o.measurable_set, inter_comm] using
     ht_int.mono_set hu_sub
 #align measure_theory.locally_integrable_on_of_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict
+-/
 
+#print MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict /-
 /-- If `s` is closed, being locally integrable on `s` wrt `μ` is equivalent to being locally
 integrable with respect to `μ.restrict s`. For the one-way implication without assuming `s` closed,
 see `locally_integrable_on_of_locally_integrable_restrict`. -/
@@ -183,13 +204,17 @@ theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace
     rw [integrable_on, restrict_restrict, inter_comm, inter_compl_self, ← integrable_on]
     exacts [integrable_on_empty, hs.measurable_set]
 #align measure_theory.locally_integrable_on_iff_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
+-/
 
+#print MeasureTheory.LocallyIntegrable.integrableOn_isCompact /-
 /-- If a function is locally integrable, then it is integrable on any compact set. -/
 theorem LocallyIntegrable.integrableOn_isCompact {k : Set X} (hf : LocallyIntegrable f μ)
     (hk : IsCompact k) : IntegrableOn f k μ :=
   (hf.LocallyIntegrableOn k).integrableOn_isCompact hk
 #align measure_theory.locally_integrable.integrable_on_is_compact MeasureTheory.LocallyIntegrable.integrableOn_isCompact
+-/
 
+#print MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact /-
 /-- If a function is locally integrable, then it is integrable on an open neighborhood of any
 compact set. -/
 theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f μ) {k : Set X}
@@ -206,19 +231,25 @@ theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f
     rcases mem_nhds_iff.1 ux with ⟨v, vu, v_open, xv⟩
     exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, subset.rfl, hu.mono_set vu⟩
 #align measure_theory.locally_integrable.integrable_on_nhds_is_compact MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact
+-/
 
+#print MeasureTheory.locallyIntegrable_iff /-
 theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact k → IntegrableOn f k μ :=
   ⟨fun hf k hk => hf.integrableOn_isCompact hk, fun hf x =>
     let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
     ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
+-/
 
+#print MeasureTheory.LocallyIntegrable.aestronglyMeasurable /-
 theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrable f μ) : AEStronglyMeasurable f μ := by
   simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AEStronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aestronglyMeasurable
+-/
 
+#print MeasureTheory.locallyIntegrable_const /-
 theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by
   intro x
@@ -226,12 +257,16 @@ theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
   refine' ⟨U, hU, _⟩
   simp only [h'U, integrable_on_const, or_true_iff]
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
+-/
 
+#print MeasureTheory.locallyIntegrableOn_const /-
 theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrableOn (fun x => c) s μ :=
   (locallyIntegrable_const c).LocallyIntegrableOn s
 #align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
+-/
 
+#print MeasureTheory.LocallyIntegrable.indicator /-
 theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
     (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ :=
   by
@@ -239,7 +274,9 @@ theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
   rcases hf x with ⟨U, hU, h'U⟩
   exact ⟨U, hU, h'U.indicator hs⟩
 #align measure_theory.locally_integrable.indicator MeasureTheory.LocallyIntegrable.indicator
+-/
 
+#print MeasureTheory.locallyIntegrable_map_homeomorph /-
 theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
     {μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ :=
   by
@@ -255,6 +292,7 @@ theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X 
     ext x
     simp only [mem_preimage, Homeomorph.symm_apply_apply]
 #align measure_theory.locally_integrable_map_homeomorph MeasureTheory.locallyIntegrable_map_homeomorph
+-/
 
 end MeasureTheory
 
@@ -266,21 +304,26 @@ variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
 
 variable {K : Set X} {a b : X}
 
+#print Continuous.locallyIntegrable /-
 /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/
 theorem Continuous.locallyIntegrable [SecondCountableTopologyEither X E] (hf : Continuous f) :
     LocallyIntegrable f μ :=
   hf.integrableAt_nhds
 #align continuous.locally_integrable Continuous.locallyIntegrable
+-/
 
+#print ContinuousOn.locallyIntegrableOn /-
 /-- A function `f` continuous on a set `K` is locally integrable on this set with respect
 to any locally finite measure. -/
 theorem ContinuousOn.locallyIntegrableOn [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
     (hK : MeasurableSet K) : LocallyIntegrableOn f K μ := fun x hx =>
   hf.integrableAt_nhdsWithin hK hx
 #align continuous_on.locally_integrable_on ContinuousOn.locallyIntegrableOn
+-/
 
 variable [MetrizableSpace X]
 
+#print ContinuousOn.integrableOn_compact /-
 /-- A function `f` continuous on a compact set `K` is integrable on this set with respect to any
 locally finite measure. -/
 theorem ContinuousOn.integrableOn_compact (hK : IsCompact K) (hf : ContinuousOn f K) :
@@ -289,43 +332,58 @@ theorem ContinuousOn.integrableOn_compact (hK : IsCompact K) (hf : ContinuousOn
   refine' locally_integrable_on.integrable_on_is_compact (fun x hx => _) hK
   exact hf.integrable_at_nhds_within_of_is_separable hK.measurable_set hK.is_separable hx
 #align continuous_on.integrable_on_compact ContinuousOn.integrableOn_compact
+-/
 
+#print ContinuousOn.integrableOn_Icc /-
 theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X]
     (hf : ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) μ :=
   hf.integrableOn_compact isCompact_Icc
 #align continuous_on.integrable_on_Icc ContinuousOn.integrableOn_Icc
+-/
 
+#print Continuous.integrableOn_Icc /-
 theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Icc a b) μ :=
   hf.ContinuousOn.integrableOn_Icc
 #align continuous.integrable_on_Icc Continuous.integrableOn_Icc
+-/
 
+#print Continuous.integrableOn_Ioc /-
 theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Ioc a b) μ :=
   hf.integrableOn_Icc.mono_set Ioc_subset_Icc_self
 #align continuous.integrable_on_Ioc Continuous.integrableOn_Ioc
+-/
 
+#print ContinuousOn.integrableOn_uIcc /-
 theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X]
     (hf : ContinuousOn f [a, b]) : IntegrableOn f [a, b] μ :=
   hf.integrableOn_Icc
 #align continuous_on.integrable_on_uIcc ContinuousOn.integrableOn_uIcc
+-/
 
+#print Continuous.integrableOn_uIcc /-
 theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f [a, b] μ :=
   hf.integrableOn_Icc
 #align continuous.integrable_on_uIcc Continuous.integrableOn_uIcc
+-/
 
+#print Continuous.integrableOn_uIoc /-
 theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Ι a b) μ :=
   hf.integrableOn_Ioc
 #align continuous.integrable_on_uIoc Continuous.integrableOn_uIoc
+-/
 
+#print Continuous.integrable_of_hasCompactSupport /-
 /-- A continuous function with compact support is integrable on the whole space. -/
 theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
     Integrable f μ :=
   (integrableOn_iff_integrable_of_support_subset (subset_tsupport f)).mp <|
     hf.ContinuousOn.integrableOn_compact hcf
 #align continuous.integrable_of_has_compact_support Continuous.integrable_of_hasCompactSupport
+-/
 
 end borel
 
@@ -336,6 +394,7 @@ section Monotone
 variable [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E]
   [OrderTopology X] [OrderTopology E] [SecondCountableTopology E]
 
+#print MonotoneOn.integrableOn_of_measure_ne_top /-
 theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
     (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
     IntegrableOn f s μ := by
@@ -352,7 +411,9 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
     integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).AEStronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
+-/
 
+#print MonotoneOn.integrableOn_isCompact /-
 theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hmono : MonotoneOn f s) : IntegrableOn f s μ :=
   by
@@ -363,18 +424,24 @@ theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : I
       hmono.integrable_on_of_measure_ne_top (hs.is_least_Inf h) (hs.is_greatest_Sup h)
         hs.measure_lt_top.ne hs.measurable_set
 #align monotone_on.integrable_on_is_compact MonotoneOn.integrableOn_isCompact
+-/
 
+#print AntitoneOn.integrableOn_of_measure_ne_top /-
 theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b : X}
     (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
     IntegrableOn f s μ :=
   hanti.dual_right.integrableOn_of_measure_ne_top ha hb hs h's
 #align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOn_of_measure_ne_top
+-/
 
+#print AntioneOn.integrableOn_isCompact /-
 theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hanti : AntitoneOn f s) : IntegrableOn f s μ :=
   hanti.dual_right.integrableOn_isCompact hs
 #align antione_on.integrable_on_is_compact AntioneOn.integrableOn_isCompact
+-/
 
+#print Monotone.locallyIntegrable /-
 theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
     LocallyIntegrable f μ := by
   intro x
@@ -387,11 +454,14 @@ theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone
     (hmono.monotone_on _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)
       ((measure_mono abU).trans_lt h'U).Ne measurableSet_Icc
 #align monotone.locally_integrable Monotone.locallyIntegrable
+-/
 
+#print Antitone.locallyIntegrable /-
 theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
     LocallyIntegrable f μ :=
   hanti.dual_right.LocallyIntegrable
 #align antitone.locally_integrable Antitone.locallyIntegrable
+-/
 
 end Monotone
 
@@ -403,6 +473,7 @@ section Mul
 
 variable [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R}
 
+#print MeasureTheory.IntegrableOn.mul_continuousOn_of_subset /-
 theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
     (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ :=
@@ -419,12 +490,16 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
     mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AEStronglyMeasurable hA)
       this
 #align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
+-/
 
+#print MeasureTheory.IntegrableOn.mul_continuousOn /-
 theorem IntegrableOn.mul_continuousOn [T2Space X] (hg : IntegrableOn g K μ)
     (hg' : ContinuousOn g' K) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
   hg.mul_continuousOn_of_subset hg' hK.MeasurableSet hK (Subset.refl _)
 #align measure_theory.integrable_on.mul_continuous_on MeasureTheory.IntegrableOn.mul_continuousOn
+-/
 
+#print MeasureTheory.IntegrableOn.continuousOn_mul_of_subset /-
 theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
     (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ :=
@@ -440,11 +515,14 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
     mem_ℒp.of_le_mul hg' (((hg.mono hAK).AEStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
       this
 #align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
+-/
 
+#print MeasureTheory.IntegrableOn.continuousOn_mul /-
 theorem IntegrableOn.continuousOn_mul [T2Space X] (hg : ContinuousOn g K)
     (hg' : IntegrableOn g' K μ) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
   hg'.continuousOn_mul_of_subset hg hK hK.MeasurableSet Subset.rfl
 #align measure_theory.integrable_on.continuous_on_mul MeasureTheory.IntegrableOn.continuousOn_mul
+-/
 
 end Mul
 
@@ -452,6 +530,7 @@ section Smul
 
 variable {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 E]
 
+#print MeasureTheory.IntegrableOn.continuousOn_smul /-
 theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
     (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
     IntegrableOn (fun x => f x • g x) K μ :=
@@ -462,7 +541,9 @@ theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEithe
     exact continuous_norm.comp_continuous_on hf
   · exact (hf.ae_strongly_measurable hK.measurable_set).smul hg.1
 #align measure_theory.integrable_on.continuous_on_smul MeasureTheory.IntegrableOn.continuousOn_smul
+-/
 
+#print MeasureTheory.IntegrableOn.smul_continuousOn /-
 theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
     (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
     IntegrableOn (fun x => f x • g x) K μ :=
@@ -473,11 +554,13 @@ theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEithe
     exact continuous_norm.comp_continuous_on hg
   · exact hf.1.smul (hg.ae_strongly_measurable hK.measurable_set)
 #align measure_theory.integrable_on.smul_continuous_on MeasureTheory.IntegrableOn.smul_continuousOn
+-/
 
 end Smul
 
 namespace LocallyIntegrableOn
 
+#print MeasureTheory.LocallyIntegrableOn.continuousOn_mul /-
 theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ :=
@@ -485,7 +568,9 @@ theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOn_mul
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.mul_continuousOn /-
 theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ :=
@@ -493,7 +578,9 @@ theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mul_continuousOn
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.continuousOn_smul /-
 theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
     [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
@@ -502,7 +589,9 @@ theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOn_smul
+-/
 
+#print MeasureTheory.LocallyIntegrableOn.smul_continuousOn /-
 theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
     [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
@@ -511,6 +600,7 @@ theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smul_continuousOn
+-/
 
 end LocallyIntegrableOn
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 
 ! This file was ported from Lean 3 source module measure_theory.function.locally_integrable
-! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
+! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -13,6 +13,9 @@ import Mathbin.MeasureTheory.Integral.IntegrableOn
 /-!
 # Locally integrable functions
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A function is called *locally integrable* (`measure_theory.locally_integrable`) if it is integrable
 on a neighborhood of every point. More generally, it is *locally integrable on `s`* if it is
 locally integrable on a neighbourhood within `s` of any point of `s`.
@@ -216,7 +219,7 @@ theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
   simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AEStronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aestronglyMeasurable
 
-theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by
   intro x
   rcases μ.finite_at_nhds x with ⟨U, hU, h'U⟩
@@ -224,7 +227,7 @@ theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
   simp only [h'U, integrable_on_const, or_true_iff]
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
 
-theorem locallyIntegrableOn_const [LocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrableOn (fun x => c) s μ :=
   (locallyIntegrable_const c).LocallyIntegrableOn s
 #align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
@@ -259,7 +262,7 @@ open MeasureTheory
 
 section borel
 
-variable [OpensMeasurableSpace X] [LocallyFiniteMeasure μ]
+variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
 
 variable {K : Set X} {a b : X}
 
@@ -350,7 +353,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 
-theorem MonotoneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hmono : MonotoneOn f s) : IntegrableOn f s μ :=
   by
   obtain rfl | h := s.eq_empty_or_nonempty
@@ -367,12 +370,12 @@ theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b
   hanti.dual_right.integrableOn_of_measure_ne_top ha hb hs h's
 #align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOn_of_measure_ne_top
 
-theorem AntioneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hanti : AntitoneOn f s) : IntegrableOn f s μ :=
   hanti.dual_right.integrableOn_isCompact hs
 #align antione_on.integrable_on_is_compact AntioneOn.integrableOn_isCompact
 
-theorem Monotone.locallyIntegrable [LocallyFiniteMeasure μ] (hmono : Monotone f) :
+theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
     LocallyIntegrable f μ := by
   intro x
   rcases μ.finite_at_nhds x with ⟨U, hU, h'U⟩
@@ -385,7 +388,7 @@ theorem Monotone.locallyIntegrable [LocallyFiniteMeasure μ] (hmono : Monotone f
       ((measure_mono abU).trans_lt h'U).Ne measurableSet_Icc
 #align monotone.locally_integrable Monotone.locallyIntegrable
 
-theorem Antitone.locallyIntegrable [LocallyFiniteMeasure μ] (hanti : Antitone f) :
+theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
     LocallyIntegrable f μ :=
   hanti.dual_right.LocallyIntegrable
 #align antitone.locally_integrable Antitone.locallyIntegrable
@@ -408,7 +411,7 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
   rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ :=
     by
-    filter_upwards [ae_restrict_mem hA]with x hx
+    filter_upwards [ae_restrict_mem hA] with x hx
     refine' (norm_mul_le _ _).trans _
     rw [mul_comm]
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
@@ -430,7 +433,7 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
   rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg' ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ :=
     by
-    filter_upwards [ae_restrict_mem hA]with x hx
+    filter_upwards [ae_restrict_mem hA] with x hx
     refine' (norm_mul_le _ _).trans _
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
   exact

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -94,7 +94,7 @@ theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
     have : s ⊆ ⋃ x : s, u x := fun y hy => mem_Union_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
     obtain ⟨T, hT_count, hT_un⟩ := is_open_Union_countable u u_open
     refine' ⟨T, hT_count, _⟩
-    rw [← hT_un, bUnion_eq_Union] at this
+    rw [← hT_un, bUnion_eq_Union] at this 
     rw [← Union_inter, eq_comm, inter_eq_right_iff_subset]
     exact this
   have : Countable T := countable_coe_iff.mpr T_count
@@ -175,10 +175,10 @@ theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace
     refine' ⟨u, hu_o.mem_nhds hu_x, _⟩
     rw [integrable_on, restrict_restrict hu_o.measurable_set]
     exact ht_int.mono_set hu_sub
-  · rw [← isOpen_compl_iff] at hs
+  · rw [← isOpen_compl_iff] at hs 
     refine' ⟨sᶜ, hs.mem_nhds h, _⟩
     rw [integrable_on, restrict_restrict, inter_comm, inter_compl_self, ← integrable_on]
-    exacts[integrable_on_empty, hs.measurable_set]
+    exacts [integrable_on_empty, hs.measurable_set]
 #align measure_theory.locally_integrable_on_iff_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
 
 /-- If a function is locally integrable, then it is integrable on any compact set. -/
@@ -405,7 +405,7 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
     IntegrableOn (fun x => g x * g' x) A μ :=
   by
   rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
-  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg⊢
+  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ :=
     by
     filter_upwards [ae_restrict_mem hA]with x hx
@@ -427,7 +427,7 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
     IntegrableOn (fun x => g x * g' x) A μ :=
   by
   rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
-  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg'⊢
+  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg' ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ :=
     by
     filter_upwards [ae_restrict_mem hA]with x hx
@@ -479,7 +479,7 @@ theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ :=
   by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOn_mul
 
@@ -487,7 +487,7 @@ theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ :=
   by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mul_continuousOn
 
@@ -496,7 +496,7 @@ theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => g x • f x) s μ :=
   by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOn_smul
 
@@ -505,7 +505,7 @@ theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => f x • g x) s μ :=
   by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smul_continuousOn
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -42,6 +42,7 @@ namespace MeasureTheory
 
 section LocallyIntegrableOn
 
+#print MeasureTheory.LocallyIntegrableOn /-
 /-- A function `f : X → E` is *locally integrable on s*, for `s ⊆ X`, if for every `x ∈ s` there is
 a neighbourhood of `x` within `s` on which `f` is integrable. (Note this is, in general, strictly
 weaker than local integrability with respect to `μ.restrict s`.) -/
@@ -49,6 +50,7 @@ def LocallyIntegrableOn (f : X → E) (s : Set X)
     (μ : Measure X := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
   ∀ x : X, x ∈ s → IntegrableAtFilter f (𝓝[s] x) μ
 #align measure_theory.locally_integrable_on MeasureTheory.LocallyIntegrableOn
+-/
 
 theorem LocallyIntegrableOn.mono (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) : LocallyIntegrableOn f t μ := fun x hx =>
@@ -77,7 +79,7 @@ theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableO
   (hf.mono hst).integrableOn_isCompact ht
 #align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
 
-theorem LocallyIntegrableOn.aEStronglyMeasurable [SecondCountableTopology X]
+theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) :=
   by
   have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ integrable_on f (u ∩ s) μ :=
@@ -98,7 +100,7 @@ theorem LocallyIntegrableOn.aEStronglyMeasurable [SecondCountableTopology X]
   have : Countable T := countable_coe_iff.mpr T_count
   rw [hT, aestronglyMeasurable_iUnion_iff]
   exact fun i : T => (hu i).AEStronglyMeasurable
-#align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aEStronglyMeasurable
+#align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable
 
 /-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
 every compact subset contained in `s`. -/
@@ -122,6 +124,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
 
 end LocallyIntegrableOn
 
+#print MeasureTheory.LocallyIntegrable /-
 /-- A function `f : X → E` is *locally integrable* if it is integrable on a neighborhood of every
 point. In particular, it is integrable on all compact sets,
 see `locally_integrable.integrable_on_is_compact`. -/
@@ -129,6 +132,7 @@ def LocallyIntegrable (f : X → E) (μ : Measure X := by exact MeasureTheory.Me
     Prop :=
   ∀ x : X, IntegrableAtFilter f (𝓝 x) μ
 #align measure_theory.locally_integrable MeasureTheory.LocallyIntegrable
+-/
 
 theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ := by
   simpa only [locally_integrable_on, nhdsWithin_univ, mem_univ, true_imp_iff]
@@ -138,9 +142,11 @@ theorem LocallyIntegrable.locallyIntegrableOn (hf : LocallyIntegrable f μ) (s :
     LocallyIntegrableOn f s μ := fun x hx => (hf x).filter_mono nhdsWithin_le_nhds
 #align measure_theory.locally_integrable.locally_integrable_on MeasureTheory.LocallyIntegrable.locallyIntegrableOn
 
+#print MeasureTheory.Integrable.locallyIntegrable /-
 theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable f μ := fun x =>
   hf.IntegrableAtFilter _
 #align measure_theory.integrable.locally_integrable MeasureTheory.Integrable.locallyIntegrable
+-/
 
 /-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
 (See `locally_integrable_on_iff_locally_integrable_restrict` for an iff statement when `s` is
@@ -205,10 +211,10 @@ theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
 
-theorem LocallyIntegrable.aEStronglyMeasurable [SecondCountableTopology X]
+theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrable f μ) : AEStronglyMeasurable f μ := by
   simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AEStronglyMeasurable
-#align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aEStronglyMeasurable
+#align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aestronglyMeasurable
 
 theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -260,14 +260,14 @@ variable {K : Set X} {a b : X}
 /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/
 theorem Continuous.locallyIntegrable [SecondCountableTopologyEither X E] (hf : Continuous f) :
     LocallyIntegrable f μ :=
-  hf.integrable_at_nhds
+  hf.integrableAt_nhds
 #align continuous.locally_integrable Continuous.locallyIntegrable
 
 /-- A function `f` continuous on a set `K` is locally integrable on this set with respect
 to any locally finite measure. -/
 theorem ContinuousOn.locallyIntegrableOn [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
     (hK : MeasurableSet K) : LocallyIntegrableOn f K μ := fun x hx =>
-  hf.integrable_at_nhdsWithin hK hx
+  hf.integrableAt_nhdsWithin hK hx
 #align continuous_on.locally_integrable_on ContinuousOn.locallyIntegrableOn
 
 variable [MetrizableSpace X]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -30,7 +30,7 @@ on compact sets.
 
 open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace
 
-open Topology Interval
+open scoped Topology Interval
 
 variable {X Y E R : Type _} [MeasurableSpace X] [TopologicalSpace X]
 
@@ -320,7 +320,7 @@ theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : Ha
 
 end borel
 
-open ENNReal
+open scoped ENNReal
 
 section Monotone
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -77,8 +77,8 @@ theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableO
   (hf.mono hst).integrableOn_isCompact ht
 #align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
 
-theorem LocallyIntegrableOn.aeStronglyMeasurable [SecondCountableTopology X]
-    (hf : LocallyIntegrableOn f s μ) : AeStronglyMeasurable f (μ.restrict s) :=
+theorem LocallyIntegrableOn.aEStronglyMeasurable [SecondCountableTopology X]
+    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) :=
   by
   have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ integrable_on f (u ∩ s) μ :=
     by
@@ -96,9 +96,9 @@ theorem LocallyIntegrableOn.aeStronglyMeasurable [SecondCountableTopology X]
     rw [← Union_inter, eq_comm, inter_eq_right_iff_subset]
     exact this
   have : Countable T := countable_coe_iff.mpr T_count
-  rw [hT, aeStronglyMeasurable_iUnion_iff]
-  exact fun i : T => (hu i).AeStronglyMeasurable
-#align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aeStronglyMeasurable
+  rw [hT, aestronglyMeasurable_iUnion_iff]
+  exact fun i : T => (hu i).AEStronglyMeasurable
+#align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aEStronglyMeasurable
 
 /-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
 every compact subset contained in `s`. -/
@@ -205,10 +205,10 @@ theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
 
-theorem LocallyIntegrable.aeStronglyMeasurable [SecondCountableTopology X]
-    (hf : LocallyIntegrable f μ) : AeStronglyMeasurable f μ := by
-  simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AeStronglyMeasurable
-#align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aeStronglyMeasurable
+theorem LocallyIntegrable.aEStronglyMeasurable [SecondCountableTopology X]
+    (hf : LocallyIntegrable f μ) : AEStronglyMeasurable f μ := by
+  simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AEStronglyMeasurable
+#align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aEStronglyMeasurable
 
 theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by
@@ -340,7 +340,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   have A : integrable_on (fun x => C) s μ := by
     simp only [hs.lt_top, integrable_on_const, or_true_iff]
   refine'
-    integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).AeStronglyMeasurable
+    integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).AEStronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 
@@ -407,7 +407,7 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
     rw [mul_comm]
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
   exact
-    mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AeStronglyMeasurable hA)
+    mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AEStronglyMeasurable hA)
       this
 #align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
 
@@ -428,7 +428,7 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
     refine' (norm_mul_le _ _).trans _
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
   exact
-    mem_ℒp.of_le_mul hg' (((hg.mono hAK).AeStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
+    mem_ℒp.of_le_mul hg' (((hg.mono hAK).AEStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
       this
 #align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -340,7 +340,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   have A : integrable_on (fun x => C) s μ := by
     simp only [hs.lt_top, integrable_on_const, or_true_iff]
   refine'
-    integrable.mono' A (aEMeasurable_restrict_of_monotoneOn h's hmono).AeStronglyMeasurable
+    integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).AeStronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -96,7 +96,7 @@ theorem LocallyIntegrableOn.aeStronglyMeasurable [SecondCountableTopology X]
     rw [← Union_inter, eq_comm, inter_eq_right_iff_subset]
     exact this
   have : Countable T := countable_coe_iff.mpr T_count
-  rw [hT, aeStronglyMeasurable_unionᵢ_iff]
+  rw [hT, aeStronglyMeasurable_iUnion_iff]
   exact fun i : T => (hu i).AeStronglyMeasurable
 #align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aeStronglyMeasurable
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -210,7 +210,7 @@ theorem LocallyIntegrable.aeStronglyMeasurable [SecondCountableTopology X]
   simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AeStronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aeStronglyMeasurable
 
-theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by
   intro x
   rcases μ.finite_at_nhds x with ⟨U, hU, h'U⟩
@@ -218,7 +218,7 @@ theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
   simp only [h'U, integrable_on_const, or_true_iff]
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
 
-theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrableOn_const [LocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrableOn (fun x => c) s μ :=
   (locallyIntegrable_const c).LocallyIntegrableOn s
 #align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
@@ -253,7 +253,7 @@ open MeasureTheory
 
 section borel
 
-variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
+variable [OpensMeasurableSpace X] [LocallyFiniteMeasure μ]
 
 variable {K : Set X} {a b : X}
 
@@ -344,7 +344,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 
-theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem MonotoneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hmono : MonotoneOn f s) : IntegrableOn f s μ :=
   by
   obtain rfl | h := s.eq_empty_or_nonempty
@@ -361,12 +361,12 @@ theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b
   hanti.dual_right.integrableOn_of_measure_ne_top ha hb hs h's
 #align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOn_of_measure_ne_top
 
-theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem AntioneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hanti : AntitoneOn f s) : IntegrableOn f s μ :=
   hanti.dual_right.integrableOn_isCompact hs
 #align antione_on.integrable_on_is_compact AntioneOn.integrableOn_isCompact
 
-theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
+theorem Monotone.locallyIntegrable [LocallyFiniteMeasure μ] (hmono : Monotone f) :
     LocallyIntegrable f μ := by
   intro x
   rcases μ.finite_at_nhds x with ⟨U, hU, h'U⟩
@@ -379,7 +379,7 @@ theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone
       ((measure_mono abU).trans_lt h'U).Ne measurableSet_Icc
 #align monotone.locally_integrable Monotone.locallyIntegrable
 
-theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
+theorem Antitone.locallyIntegrable [LocallyFiniteMeasure μ] (hanti : Antitone f) :
     LocallyIntegrable f μ :=
   hanti.dual_right.LocallyIntegrable
 #align antitone.locally_integrable Antitone.locallyIntegrable

mathlib commit https://github.com/leanprover-community/mathlib/commit/92c69b77c5a7dc0f7eeddb552508633305157caa

@@ -66,16 +66,16 @@ theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyInt
 #align measure_theory.integrable_on.locally_integrable_on MeasureTheory.IntegrableOn.locallyIntegrableOn
 
 /-- If a function is locally integrable on a compact set, then it is integrable on that set. -/
-theorem LocallyIntegrableOn.integrableOnIsCompact (hf : LocallyIntegrableOn f s μ)
+theorem LocallyIntegrableOn.integrableOn_isCompact (hf : LocallyIntegrableOn f s μ)
     (hs : IsCompact s) : IntegrableOn f s μ :=
-  IsCompact.induction_on hs integrableOnEmpty (fun u v huv hv => hv.monoSet huv)
+  IsCompact.induction_on hs integrableOn_empty (fun u v huv hv => hv.mono_set huv)
     (fun u v hu hv => integrableOn_union.mpr ⟨hu, hv⟩) hf
-#align measure_theory.locally_integrable_on.integrable_on_is_compact MeasureTheory.LocallyIntegrableOn.integrableOnIsCompact
+#align measure_theory.locally_integrable_on.integrable_on_is_compact MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact
 
-theorem LocallyIntegrableOn.integrableOnCompactSubset (hf : LocallyIntegrableOn f s μ) {t : Set X}
+theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t μ :=
-  (hf.mono hst).integrableOnIsCompact ht
-#align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOnCompactSubset
+  (hf.mono hst).integrableOn_isCompact ht
+#align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
 
 theorem LocallyIntegrableOn.aeStronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrableOn f s μ) : AeStronglyMeasurable f (μ.restrict s) :=
@@ -107,7 +107,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
   by
   -- The correct condition is that `s` be *locally closed*, i.e. for every `x ∈ s` there is some
   -- `U ∈ 𝓝 x` such that `U ∩ s` is closed. But mathlib doesn't have locally closed sets yet.
-  refine' ⟨fun hf k hk => hf.integrableOnCompactSubset hk, fun hf x hx => _⟩
+  refine' ⟨fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => _⟩
   cases hs
   ·
     exact
@@ -145,7 +145,7 @@ theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable
 /-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
 (See `locally_integrable_on_iff_locally_integrable_restrict` for an iff statement when `s` is
 closed.) -/
-theorem locallyIntegrableOnOfLocallyIntegrableRestrict [OpensMeasurableSpace X]
+theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace X]
     (hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn f s μ :=
   by
   intro x hx
@@ -154,7 +154,7 @@ theorem locallyIntegrableOnOfLocallyIntegrableRestrict [OpensMeasurableSpace X]
   refine' ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _⟩
   simpa only [integrable_on, measure.restrict_restrict hu_o.measurable_set, inter_comm] using
     ht_int.mono_set hu_sub
-#align measure_theory.locally_integrable_on_of_locally_integrable_restrict MeasureTheory.locallyIntegrableOnOfLocallyIntegrableRestrict
+#align measure_theory.locally_integrable_on_of_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict
 
 /-- If `s` is closed, being locally integrable on `s` wrt `μ` is equivalent to being locally
 integrable with respect to `μ.restrict s`. For the one-way implication without assuming `s` closed,
@@ -176,10 +176,10 @@ theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace
 #align measure_theory.locally_integrable_on_iff_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
 
 /-- If a function is locally integrable, then it is integrable on any compact set. -/
-theorem LocallyIntegrable.integrableOnIsCompact {k : Set X} (hf : LocallyIntegrable f μ)
+theorem LocallyIntegrable.integrableOn_isCompact {k : Set X} (hf : LocallyIntegrable f μ)
     (hk : IsCompact k) : IntegrableOn f k μ :=
-  (hf.LocallyIntegrableOn k).integrableOnIsCompact hk
-#align measure_theory.locally_integrable.integrable_on_is_compact MeasureTheory.LocallyIntegrable.integrableOnIsCompact
+  (hf.LocallyIntegrableOn k).integrableOn_isCompact hk
+#align measure_theory.locally_integrable.integrable_on_is_compact MeasureTheory.LocallyIntegrable.integrableOn_isCompact
 
 /-- If a function is locally integrable, then it is integrable on an open neighborhood of any
 compact set. -/
@@ -200,7 +200,7 @@ theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f
 
 theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact k → IntegrableOn f k μ :=
-  ⟨fun hf k hk => hf.integrableOnIsCompact hk, fun hf x =>
+  ⟨fun hf k hk => hf.integrableOn_isCompact hk, fun hf x =>
     let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
     ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
@@ -210,18 +210,18 @@ theorem LocallyIntegrable.aeStronglyMeasurable [SecondCountableTopology X]
   simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AeStronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aeStronglyMeasurable
 
-theorem locallyIntegrableConst [IsLocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun x => c) μ := by
   intro x
   rcases μ.finite_at_nhds x with ⟨U, hU, h'U⟩
   refine' ⟨U, hU, _⟩
   simp only [h'U, integrable_on_const, or_true_iff]
-#align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrableConst
+#align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
 
-theorem locallyIntegrableOnConst [IsLocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrableOn (fun x => c) s μ :=
-  (locallyIntegrableConst c).LocallyIntegrableOn s
-#align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOnConst
+  (locallyIntegrable_const c).LocallyIntegrableOn s
+#align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
 
 theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
     (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ :=
@@ -260,63 +260,63 @@ variable {K : Set X} {a b : X}
 /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/
 theorem Continuous.locallyIntegrable [SecondCountableTopologyEither X E] (hf : Continuous f) :
     LocallyIntegrable f μ :=
-  hf.integrableAtNhds
+  hf.integrable_at_nhds
 #align continuous.locally_integrable Continuous.locallyIntegrable
 
 /-- A function `f` continuous on a set `K` is locally integrable on this set with respect
 to any locally finite measure. -/
 theorem ContinuousOn.locallyIntegrableOn [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
     (hK : MeasurableSet K) : LocallyIntegrableOn f K μ := fun x hx =>
-  hf.integrableAtNhdsWithin hK hx
+  hf.integrable_at_nhdsWithin hK hx
 #align continuous_on.locally_integrable_on ContinuousOn.locallyIntegrableOn
 
 variable [MetrizableSpace X]
 
 /-- A function `f` continuous on a compact set `K` is integrable on this set with respect to any
 locally finite measure. -/
-theorem ContinuousOn.integrableOnCompact (hK : IsCompact K) (hf : ContinuousOn f K) :
+theorem ContinuousOn.integrableOn_compact (hK : IsCompact K) (hf : ContinuousOn f K) :
     IntegrableOn f K μ := by
   letI := metrizable_space_metric X
   refine' locally_integrable_on.integrable_on_is_compact (fun x hx => _) hK
   exact hf.integrable_at_nhds_within_of_is_separable hK.measurable_set hK.is_separable hx
-#align continuous_on.integrable_on_compact ContinuousOn.integrableOnCompact
+#align continuous_on.integrable_on_compact ContinuousOn.integrableOn_compact
 
-theorem ContinuousOn.integrableOnIcc [Preorder X] [CompactIccSpace X]
+theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X]
     (hf : ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) μ :=
-  hf.integrableOnCompact isCompact_Icc
-#align continuous_on.integrable_on_Icc ContinuousOn.integrableOnIcc
+  hf.integrableOn_compact isCompact_Icc
+#align continuous_on.integrable_on_Icc ContinuousOn.integrableOn_Icc
 
-theorem Continuous.integrableOnIcc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
+theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Icc a b) μ :=
-  hf.ContinuousOn.integrableOnIcc
-#align continuous.integrable_on_Icc Continuous.integrableOnIcc
+  hf.ContinuousOn.integrableOn_Icc
+#align continuous.integrable_on_Icc Continuous.integrableOn_Icc
 
-theorem Continuous.integrableOnIoc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
+theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Ioc a b) μ :=
-  hf.integrableOnIcc.monoSet Ioc_subset_Icc_self
-#align continuous.integrable_on_Ioc Continuous.integrableOnIoc
+  hf.integrableOn_Icc.mono_set Ioc_subset_Icc_self
+#align continuous.integrable_on_Ioc Continuous.integrableOn_Ioc
 
-theorem ContinuousOn.integrableOnUIcc [LinearOrder X] [CompactIccSpace X]
+theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X]
     (hf : ContinuousOn f [a, b]) : IntegrableOn f [a, b] μ :=
-  hf.integrableOnIcc
-#align continuous_on.integrable_on_uIcc ContinuousOn.integrableOnUIcc
+  hf.integrableOn_Icc
+#align continuous_on.integrable_on_uIcc ContinuousOn.integrableOn_uIcc
 
-theorem Continuous.integrableOnUIcc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
+theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f [a, b] μ :=
-  hf.integrableOnIcc
-#align continuous.integrable_on_uIcc Continuous.integrableOnUIcc
+  hf.integrableOn_Icc
+#align continuous.integrable_on_uIcc Continuous.integrableOn_uIcc
 
-theorem Continuous.integrableOnUIoc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
+theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
     IntegrableOn f (Ι a b) μ :=
-  hf.integrableOnIoc
-#align continuous.integrable_on_uIoc Continuous.integrableOnUIoc
+  hf.integrableOn_Ioc
+#align continuous.integrable_on_uIoc Continuous.integrableOn_uIoc
 
 /-- A continuous function with compact support is integrable on the whole space. -/
-theorem Continuous.integrableOfHasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
+theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
     Integrable f μ :=
   (integrableOn_iff_integrable_of_support_subset (subset_tsupport f)).mp <|
-    hf.ContinuousOn.integrableOnCompact hcf
-#align continuous.integrable_of_has_compact_support Continuous.integrableOfHasCompactSupport
+    hf.ContinuousOn.integrableOn_compact hcf
+#align continuous.integrable_of_has_compact_support Continuous.integrable_of_hasCompactSupport
 
 end borel
 
@@ -327,9 +327,9 @@ section Monotone
 variable [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E]
   [OrderTopology X] [OrderTopology E] [SecondCountableTopology E]
 
-theorem MonotoneOn.integrableOnOfMeasureNeTop (hmono : MonotoneOn f s) {a b : X} (ha : IsLeast s a)
-    (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) : IntegrableOn f s μ :=
-  by
+theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
+    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
+    IntegrableOn f s μ := by
   borelize E
   obtain rfl | h := s.eq_empty_or_nonempty
   · exact integrable_on_empty
@@ -340,11 +340,11 @@ theorem MonotoneOn.integrableOnOfMeasureNeTop (hmono : MonotoneOn f s) {a b : X}
   have A : integrable_on (fun x => C) s μ := by
     simp only [hs.lt_top, integrable_on_const, or_true_iff]
   refine'
-    integrable.mono' A (aeMeasurableRestrictOfMonotoneOn h's hmono).AeStronglyMeasurable
+    integrable.mono' A (aEMeasurable_restrict_of_monotoneOn h's hmono).AeStronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
-#align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOnOfMeasureNeTop
+#align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 
-theorem MonotoneOn.integrableOnIsCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hmono : MonotoneOn f s) : IntegrableOn f s μ :=
   by
   obtain rfl | h := s.eq_empty_or_nonempty
@@ -353,17 +353,18 @@ theorem MonotoneOn.integrableOnIsCompact [IsFiniteMeasureOnCompacts μ] (hs : Is
     exact
       hmono.integrable_on_of_measure_ne_top (hs.is_least_Inf h) (hs.is_greatest_Sup h)
         hs.measure_lt_top.ne hs.measurable_set
-#align monotone_on.integrable_on_is_compact MonotoneOn.integrableOnIsCompact
+#align monotone_on.integrable_on_is_compact MonotoneOn.integrableOn_isCompact
 
-theorem AntitoneOn.integrableOnOfMeasureNeTop (hanti : AntitoneOn f s) {a b : X} (ha : IsLeast s a)
-    (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) : IntegrableOn f s μ :=
-  hanti.dual_right.integrableOnOfMeasureNeTop ha hb hs h's
-#align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOnOfMeasureNeTop
+theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b : X}
+    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
+    IntegrableOn f s μ :=
+  hanti.dual_right.integrableOn_of_measure_ne_top ha hb hs h's
+#align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOn_of_measure_ne_top
 
-theorem AntioneOn.integrableOnIsCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hanti : AntitoneOn f s) : IntegrableOn f s μ :=
-  hanti.dual_right.integrableOnIsCompact hs
-#align antione_on.integrable_on_is_compact AntioneOn.integrableOnIsCompact
+  hanti.dual_right.integrableOn_isCompact hs
+#align antione_on.integrable_on_is_compact AntioneOn.integrableOn_isCompact
 
 theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
     LocallyIntegrable f μ := by
@@ -374,7 +375,7 @@ theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone
   have ab : a ≤ b := xab.1.trans xab.2
   refine' ⟨Icc a b, hab, _⟩
   exact
-    (hmono.monotone_on _).integrableOnOfMeasureNeTop (isLeast_Icc ab) (isGreatest_Icc ab)
+    (hmono.monotone_on _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)
       ((measure_mono abU).trans_lt h'U).Ne measurableSet_Icc
 #align monotone.locally_integrable Monotone.locallyIntegrable
 
@@ -393,7 +394,7 @@ section Mul
 
 variable [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R}
 
-theorem IntegrableOn.mulContinuousOnOfSubset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
+theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
     (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ :=
   by
@@ -408,14 +409,14 @@ theorem IntegrableOn.mulContinuousOnOfSubset (hg : IntegrableOn g A μ) (hg' : C
   exact
     mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AeStronglyMeasurable hA)
       this
-#align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mulContinuousOnOfSubset
+#align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
 
-theorem IntegrableOn.mulContinuousOn [T2Space X] (hg : IntegrableOn g K μ) (hg' : ContinuousOn g' K)
-    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
-  hg.mulContinuousOnOfSubset hg' hK.MeasurableSet hK (Subset.refl _)
-#align measure_theory.integrable_on.mul_continuous_on MeasureTheory.IntegrableOn.mulContinuousOn
+theorem IntegrableOn.mul_continuousOn [T2Space X] (hg : IntegrableOn g K μ)
+    (hg' : ContinuousOn g' K) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
+  hg.mul_continuousOn_of_subset hg' hK.MeasurableSet hK (Subset.refl _)
+#align measure_theory.integrable_on.mul_continuous_on MeasureTheory.IntegrableOn.mul_continuousOn
 
-theorem IntegrableOn.continuousOnMulOfSubset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
+theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
     (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ :=
   by
@@ -429,12 +430,12 @@ theorem IntegrableOn.continuousOnMulOfSubset (hg : ContinuousOn g K) (hg' : Inte
   exact
     mem_ℒp.of_le_mul hg' (((hg.mono hAK).AeStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
       this
-#align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOnMulOfSubset
+#align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
 
-theorem IntegrableOn.continuousOnMul [T2Space X] (hg : ContinuousOn g K) (hg' : IntegrableOn g' K μ)
-    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
-  hg'.continuousOnMulOfSubset hg hK hK.MeasurableSet Subset.rfl
-#align measure_theory.integrable_on.continuous_on_mul MeasureTheory.IntegrableOn.continuousOnMul
+theorem IntegrableOn.continuousOn_mul [T2Space X] (hg : ContinuousOn g K)
+    (hg' : IntegrableOn g' K μ) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
+  hg'.continuousOn_mul_of_subset hg hK hK.MeasurableSet Subset.rfl
+#align measure_theory.integrable_on.continuous_on_mul MeasureTheory.IntegrableOn.continuousOn_mul
 
 end Mul
 
@@ -442,7 +443,7 @@ section Smul
 
 variable {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 E]
 
-theorem IntegrableOn.continuousOnSmul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
+theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
     (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
     IntegrableOn (fun x => f x • g x) K μ :=
   by
@@ -451,9 +452,9 @@ theorem IntegrableOn.continuousOnSmul [T2Space X] [SecondCountableTopologyEither
     refine' integrable_on.continuous_on_mul _ hg.norm hK
     exact continuous_norm.comp_continuous_on hf
   · exact (hf.ae_strongly_measurable hK.measurable_set).smul hg.1
-#align measure_theory.integrable_on.continuous_on_smul MeasureTheory.IntegrableOn.continuousOnSmul
+#align measure_theory.integrable_on.continuous_on_smul MeasureTheory.IntegrableOn.continuousOn_smul
 
-theorem IntegrableOn.smulContinuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
+theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
     (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
     IntegrableOn (fun x => f x • g x) K μ :=
   by
@@ -462,45 +463,45 @@ theorem IntegrableOn.smulContinuousOn [T2Space X] [SecondCountableTopologyEither
     refine' integrable_on.mul_continuous_on hf.norm _ hK
     exact continuous_norm.comp_continuous_on hg
   · exact hf.1.smul (hg.ae_strongly_measurable hK.measurable_set)
-#align measure_theory.integrable_on.smul_continuous_on MeasureTheory.IntegrableOn.smulContinuousOn
+#align measure_theory.integrable_on.smul_continuous_on MeasureTheory.IntegrableOn.smul_continuousOn
 
 end Smul
 
 namespace LocallyIntegrableOn
 
-theorem continuousOnMul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
+theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ :=
   by
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
-  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOnMul (hg.mono hk_sub) hk_c
-#align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOnMul
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOn_mul
 
-theorem mulContinuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
+theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ :=
   by
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
-  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mulContinuousOn (hg.mono hk_sub) hk_c
-#align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mulContinuousOn
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mul_continuousOn
 
-theorem continuousOnSmul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
     [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => g x • f x) s μ :=
   by
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
-  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOnSmul (hg.mono hk_sub) hk_c
-#align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOnSmul
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOn_smul
 
-theorem smulContinuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
     [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => f x • g x) s μ :=
   by
   rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
-  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smulContinuousOn (hg.mono hk_sub) hk_c
-#align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smulContinuousOn
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smul_continuousOn
 
 end LocallyIntegrableOn
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 
 ! This file was ported from Lean 3 source module measure_theory.function.locally_integrable
-! leanprover-community/mathlib commit a75898643b2d774cced9ae7c0b28c21663b99666
+! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,15 +14,17 @@ import Mathbin.MeasureTheory.Integral.IntegrableOn
 # Locally integrable functions
 
 A function is called *locally integrable* (`measure_theory.locally_integrable`) if it is integrable
-on a neighborhood of every point.
+on a neighborhood of every point. More generally, it is *locally integrable on `s`* if it is
+locally integrable on a neighbourhood within `s` of any point of `s`.
 
-This file contains properties of locally integrable functions and integrability results
+This file contains properties of locally integrable functions, and integrability results
 on compact sets.
 
 ## Main statements
 
 * `continuous.locally_integrable`: A continuous function is locally integrable.
-
+* `continuous_on.locally_integrable_on`: A function which is continuous on `s` is locally
+  integrable on `s`.
 -/
 
 
@@ -34,11 +36,93 @@ variable {X Y E R : Type _} [MeasurableSpace X] [TopologicalSpace X]
 
 variable [MeasurableSpace Y] [TopologicalSpace Y]
 
-variable [NormedAddCommGroup E] {f : X → E} {μ : Measure X}
+variable [NormedAddCommGroup E] {f : X → E} {μ : Measure X} {s : Set X}
 
 namespace MeasureTheory
 
-/-- A function `f : X → E` is locally integrable if it is integrable on a neighborhood of every
+section LocallyIntegrableOn
+
+/-- A function `f : X → E` is *locally integrable on s*, for `s ⊆ X`, if for every `x ∈ s` there is
+a neighbourhood of `x` within `s` on which `f` is integrable. (Note this is, in general, strictly
+weaker than local integrability with respect to `μ.restrict s`.) -/
+def LocallyIntegrableOn (f : X → E) (s : Set X)
+    (μ : Measure X := by exact MeasureTheory.MeasureSpace.volume) : Prop :=
+  ∀ x : X, x ∈ s → IntegrableAtFilter f (𝓝[s] x) μ
+#align measure_theory.locally_integrable_on MeasureTheory.LocallyIntegrableOn
+
+theorem LocallyIntegrableOn.mono (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
+    (hst : t ⊆ s) : LocallyIntegrableOn f t μ := fun x hx =>
+  (hf x <| hst hx).filter_mono (nhdsWithin_mono x hst)
+#align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono
+
+theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
+    LocallyIntegrableOn (fun x => ‖f x‖) s μ := fun t ht =>
+  let ⟨U, hU_nhd, hU_int⟩ := hf t ht
+  ⟨U, hU_nhd, hU_int.norm⟩
+#align measure_theory.locally_integrable_on.norm MeasureTheory.LocallyIntegrableOn.norm
+
+theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyIntegrableOn f s μ :=
+  fun x hx => ⟨s, self_mem_nhdsWithin, hf⟩
+#align measure_theory.integrable_on.locally_integrable_on MeasureTheory.IntegrableOn.locallyIntegrableOn
+
+/-- If a function is locally integrable on a compact set, then it is integrable on that set. -/
+theorem LocallyIntegrableOn.integrableOnIsCompact (hf : LocallyIntegrableOn f s μ)
+    (hs : IsCompact s) : IntegrableOn f s μ :=
+  IsCompact.induction_on hs integrableOnEmpty (fun u v huv hv => hv.monoSet huv)
+    (fun u v hu hv => integrableOn_union.mpr ⟨hu, hv⟩) hf
+#align measure_theory.locally_integrable_on.integrable_on_is_compact MeasureTheory.LocallyIntegrableOn.integrableOnIsCompact
+
+theorem LocallyIntegrableOn.integrableOnCompactSubset (hf : LocallyIntegrableOn f s μ) {t : Set X}
+    (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t μ :=
+  (hf.mono hst).integrableOnIsCompact ht
+#align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOnCompactSubset
+
+theorem LocallyIntegrableOn.aeStronglyMeasurable [SecondCountableTopology X]
+    (hf : LocallyIntegrableOn f s μ) : AeStronglyMeasurable f (μ.restrict s) :=
+  by
+  have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ integrable_on f (u ∩ s) μ :=
+    by
+    rintro ⟨x, hx⟩
+    rcases hf x hx with ⟨t, ht, h't⟩
+    rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
+    refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
+  choose u u_open xu hu using this
+  obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s = ⋃ i : T, u i ∩ s :=
+    by
+    have : s ⊆ ⋃ x : s, u x := fun y hy => mem_Union_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
+    obtain ⟨T, hT_count, hT_un⟩ := is_open_Union_countable u u_open
+    refine' ⟨T, hT_count, _⟩
+    rw [← hT_un, bUnion_eq_Union] at this
+    rw [← Union_inter, eq_comm, inter_eq_right_iff_subset]
+    exact this
+  have : Countable T := countable_coe_iff.mpr T_count
+  rw [hT, aeStronglyMeasurable_unionᵢ_iff]
+  exact fun i : T => (hu i).AeStronglyMeasurable
+#align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aeStronglyMeasurable
+
+/-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
+every compact subset contained in `s`. -/
+theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
+    LocallyIntegrableOn f s μ ↔ ∀ (k : Set X) (hk : k ⊆ s), IsCompact k → IntegrableOn f k μ :=
+  by
+  -- The correct condition is that `s` be *locally closed*, i.e. for every `x ∈ s` there is some
+  -- `U ∈ 𝓝 x` such that `U ∩ s` is closed. But mathlib doesn't have locally closed sets yet.
+  refine' ⟨fun hf k hk => hf.integrableOnCompactSubset hk, fun hf x hx => _⟩
+  cases hs
+  ·
+    exact
+      let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
+      ⟨_, inter_mem_nhdsWithin s h2K,
+        hf _ (inter_subset_left _ _)
+          (isCompact_of_isClosed_subset hK (hs.inter hK.IsClosed) (inter_subset_right _ _))⟩
+  · obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
+    refine' ⟨K, _, hf K h3K hK⟩
+    simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
+#align measure_theory.locally_integrable_on_iff MeasureTheory.locallyIntegrableOn_iff
+
+end LocallyIntegrableOn
+
+/-- A function `f : X → E` is *locally integrable* if it is integrable on a neighborhood of every
 point. In particular, it is integrable on all compact sets,
 see `locally_integrable.integrable_on_is_compact`. -/
 def LocallyIntegrable (f : X → E) (μ : Measure X := by exact MeasureTheory.MeasureSpace.volume) :
@@ -46,10 +130,57 @@ def LocallyIntegrable (f : X → E) (μ : Measure X := by exact MeasureTheory.Me
   ∀ x : X, IntegrableAtFilter f (𝓝 x) μ
 #align measure_theory.locally_integrable MeasureTheory.LocallyIntegrable
 
+theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ := by
+  simpa only [locally_integrable_on, nhdsWithin_univ, mem_univ, true_imp_iff]
+#align measure_theory.locally_integrable_on_univ MeasureTheory.locallyIntegrableOn_univ
+
+theorem LocallyIntegrable.locallyIntegrableOn (hf : LocallyIntegrable f μ) (s : Set X) :
+    LocallyIntegrableOn f s μ := fun x hx => (hf x).filter_mono nhdsWithin_le_nhds
+#align measure_theory.locally_integrable.locally_integrable_on MeasureTheory.LocallyIntegrable.locallyIntegrableOn
+
 theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable f μ := fun x =>
   hf.IntegrableAtFilter _
 #align measure_theory.integrable.locally_integrable MeasureTheory.Integrable.locallyIntegrable
 
+/-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
+(See `locally_integrable_on_iff_locally_integrable_restrict` for an iff statement when `s` is
+closed.) -/
+theorem locallyIntegrableOnOfLocallyIntegrableRestrict [OpensMeasurableSpace X]
+    (hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn f s μ :=
+  by
+  intro x hx
+  obtain ⟨t, ht_mem, ht_int⟩ := hf x
+  obtain ⟨u, hu_sub, hu_o, hu_mem⟩ := mem_nhds_iff.mp ht_mem
+  refine' ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _⟩
+  simpa only [integrable_on, measure.restrict_restrict hu_o.measurable_set, inter_comm] using
+    ht_int.mono_set hu_sub
+#align measure_theory.locally_integrable_on_of_locally_integrable_restrict MeasureTheory.locallyIntegrableOnOfLocallyIntegrableRestrict
+
+/-- If `s` is closed, being locally integrable on `s` wrt `μ` is equivalent to being locally
+integrable with respect to `μ.restrict s`. For the one-way implication without assuming `s` closed,
+see `locally_integrable_on_of_locally_integrable_restrict`. -/
+theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace X]
+    (hs : IsClosed s) : LocallyIntegrableOn f s μ ↔ LocallyIntegrable f (μ.restrict s) :=
+  by
+  refine' ⟨fun hf x => _, locally_integrable_on_of_locally_integrable_restrict⟩
+  by_cases h : x ∈ s
+  · obtain ⟨t, ht_nhds, ht_int⟩ := hf x h
+    obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhds_within.mp ht_nhds
+    refine' ⟨u, hu_o.mem_nhds hu_x, _⟩
+    rw [integrable_on, restrict_restrict hu_o.measurable_set]
+    exact ht_int.mono_set hu_sub
+  · rw [← isOpen_compl_iff] at hs
+    refine' ⟨sᶜ, hs.mem_nhds h, _⟩
+    rw [integrable_on, restrict_restrict, inter_comm, inter_compl_self, ← integrable_on]
+    exacts[integrable_on_empty, hs.measurable_set]
+#align measure_theory.locally_integrable_on_iff_locally_integrable_restrict MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
+
+/-- If a function is locally integrable, then it is integrable on any compact set. -/
+theorem LocallyIntegrable.integrableOnIsCompact {k : Set X} (hf : LocallyIntegrable f μ)
+    (hk : IsCompact k) : IntegrableOn f k μ :=
+  (hf.LocallyIntegrableOn k).integrableOnIsCompact hk
+#align measure_theory.locally_integrable.integrable_on_is_compact MeasureTheory.LocallyIntegrable.integrableOnIsCompact
+
 /-- If a function is locally integrable, then it is integrable on an open neighborhood of any
 compact set. -/
 theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f μ) {k : Set X}
@@ -67,41 +198,16 @@ theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f
     exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, subset.rfl, hu.mono_set vu⟩
 #align measure_theory.locally_integrable.integrable_on_nhds_is_compact MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact
 
-/-- If a function is locally integrable, then it is integrable on any compact set. -/
-theorem LocallyIntegrable.integrableOnIsCompact {k : Set X} (hf : LocallyIntegrable f μ)
-    (hk : IsCompact k) : IntegrableOn f k μ :=
-  by
-  rcases hf.integrable_on_nhds_is_compact hk with ⟨u, u_open, ku, hu⟩
-  exact hu.mono_set ku
-#align measure_theory.locally_integrable.integrable_on_is_compact MeasureTheory.LocallyIntegrable.integrableOnIsCompact
-
 theorem locallyIntegrable_iff [LocallyCompactSpace X] :
     LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact k → IntegrableOn f k μ :=
-  by
-  refine' ⟨fun hf k hk => hf.integrableOnIsCompact hk, fun hf x => _⟩
-  obtain ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
-  exact ⟨K, h2K, hf K hK⟩
+  ⟨fun hf k hk => hf.integrableOnIsCompact hk, fun hf x =>
+    let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
+    ⟨K, h2K, hf K hK⟩⟩
 #align measure_theory.locally_integrable_iff MeasureTheory.locallyIntegrable_iff
 
 theorem LocallyIntegrable.aeStronglyMeasurable [SecondCountableTopology X]
-    (hf : LocallyIntegrable f μ) : AeStronglyMeasurable f μ :=
-  by
-  have : ∀ x, ∃ u, IsOpen u ∧ x ∈ u ∧ integrable_on f u μ :=
-    by
-    intro x
-    rcases hf x with ⟨s, hs, h's⟩
-    rcases mem_nhds_iff.1 hs with ⟨u, us, u_open, xu⟩
-    exact ⟨u, u_open, xu, h's.mono_set us⟩
-  choose u u_open xu hu using this
-  obtain ⟨T, T_count, hT⟩ : ∃ T : Set X, T.Countable ∧ (⋃ i : T, u i) = univ :=
-    by
-    have : (⋃ x, u x) = univ := eq_univ_of_forall fun x => mem_Union_of_mem x (xu x)
-    rw [← this]
-    simp only [Union_coe_set, Subtype.coe_mk]
-    exact is_open_Union_countable u u_open
-  have : Countable T := countable_coe_iff.mpr T_count
-  rw [← @restrict_univ _ _ μ, ← hT, aeStronglyMeasurable_unionᵢ_iff]
-  exact fun i => (hu i).AeStronglyMeasurable
+    (hf : LocallyIntegrable f μ) : AeStronglyMeasurable f μ := by
+  simpa only [restrict_univ] using (locally_integrable_on_univ.mpr hf).AeStronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aeStronglyMeasurable
 
 theorem locallyIntegrableConst [IsLocallyFiniteMeasure μ] (c : E) :
@@ -112,6 +218,11 @@ theorem locallyIntegrableConst [IsLocallyFiniteMeasure μ] (c : E) :
   simp only [h'U, integrable_on_const, or_true_iff]
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrableConst
 
+theorem locallyIntegrableOnConst [IsLocallyFiniteMeasure μ] (c : E) :
+    LocallyIntegrableOn (fun x => c) s μ :=
+  (locallyIntegrableConst c).LocallyIntegrableOn s
+#align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOnConst
+
 theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
     (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ :=
   by
@@ -136,68 +247,10 @@ theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X 
     simp only [mem_preimage, Homeomorph.symm_apply_apply]
 #align measure_theory.locally_integrable_map_homeomorph MeasureTheory.locallyIntegrable_map_homeomorph
 
-section Mul
-
-variable [OpensMeasurableSpace X] [NormedRing R] [SecondCountableTopologyEither X R] {A K : Set X}
-  {g g' : X → R}
-
-theorem IntegrableOn.mulContinuousOnOfSubset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
-    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
-    IntegrableOn (fun x => g x * g' x) A μ :=
-  by
-  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
-  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg⊢
-  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ :=
-    by
-    filter_upwards [ae_restrict_mem hA]with x hx
-    refine' (norm_mul_le _ _).trans _
-    rw [mul_comm]
-    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
-  exact
-    mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AeStronglyMeasurable hA)
-      this
-#align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mulContinuousOnOfSubset
-
-theorem IntegrableOn.mulContinuousOn [T2Space X] (hg : IntegrableOn g K μ) (hg' : ContinuousOn g' K)
-    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
-  hg.mulContinuousOnOfSubset hg' hK.MeasurableSet hK (Subset.refl _)
-#align measure_theory.integrable_on.mul_continuous_on MeasureTheory.IntegrableOn.mulContinuousOn
-
-theorem IntegrableOn.continuousOnMulOfSubset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
-    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
-    IntegrableOn (fun x => g x * g' x) A μ :=
-  by
-  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
-  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg'⊢
-  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ :=
-    by
-    filter_upwards [ae_restrict_mem hA]with x hx
-    refine' (norm_mul_le _ _).trans _
-    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
-  exact
-    mem_ℒp.of_le_mul hg' (((hg.mono hAK).AeStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
-      this
-#align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOnMulOfSubset
-
-theorem IntegrableOn.continuousOnMul [T2Space X] (hg : ContinuousOn g K) (hg' : IntegrableOn g' K μ)
-    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
-  hg'.continuousOnMulOfSubset hg hK hK.MeasurableSet Subset.rfl
-#align measure_theory.integrable_on.continuous_on_mul MeasureTheory.IntegrableOn.continuousOnMul
-
-end Mul
-
 end MeasureTheory
 
 open MeasureTheory
 
-/-- If a function is integrable at `𝓝[s] x` for each point `x` of a compact set `s`, then it is
-integrable on `s`. -/
-theorem IsCompact.integrableOnOfNhdsWithin {K : Set X} (hK : IsCompact K)
-    (hf : ∀ x ∈ K, IntegrableAtFilter f (𝓝[K] x) μ) : IntegrableOn f K μ :=
-  IsCompact.induction_on hK integrableOnEmpty (fun s t hst ht => ht.monoSet hst)
-    (fun s t hs ht => hs.union ht) hf
-#align is_compact.integrable_on_of_nhds_within IsCompact.integrableOnOfNhdsWithin
-
 section borel
 
 variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
@@ -210,6 +263,13 @@ theorem Continuous.locallyIntegrable [SecondCountableTopologyEither X E] (hf : C
   hf.integrableAtNhds
 #align continuous.locally_integrable Continuous.locallyIntegrable
 
+/-- A function `f` continuous on a set `K` is locally integrable on this set with respect
+to any locally finite measure. -/
+theorem ContinuousOn.locallyIntegrableOn [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
+    (hK : MeasurableSet K) : LocallyIntegrableOn f K μ := fun x hx =>
+  hf.integrableAtNhdsWithin hK hx
+#align continuous_on.locally_integrable_on ContinuousOn.locallyIntegrableOn
+
 variable [MetrizableSpace X]
 
 /-- A function `f` continuous on a compact set `K` is integrable on this set with respect to any
@@ -217,7 +277,7 @@ locally finite measure. -/
 theorem ContinuousOn.integrableOnCompact (hK : IsCompact K) (hf : ContinuousOn f K) :
     IntegrableOn f K μ := by
   letI := metrizable_space_metric X
-  apply hK.integrable_on_of_nhds_within fun x hx => _
+  refine' locally_integrable_on.integrable_on_is_compact (fun x hx => _) hK
   exact hf.integrable_at_nhds_within_of_is_separable hK.measurable_set hK.is_separable hx
 #align continuous_on.integrable_on_compact ContinuousOn.integrableOnCompact
 
@@ -265,7 +325,7 @@ open ENNReal
 section Monotone
 
 variable [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E]
-  [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] {s : Set X}
+  [OrderTopology X] [OrderTopology E] [SecondCountableTopology E]
 
 theorem MonotoneOn.integrableOnOfMeasureNeTop (hmono : MonotoneOn f s) {a b : X} (ha : IsLeast s a)
     (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) : IntegrableOn f s μ :=
@@ -325,3 +385,124 @@ theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone
 
 end Monotone
 
+namespace MeasureTheory
+
+variable [OpensMeasurableSpace X] {A K : Set X}
+
+section Mul
+
+variable [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R}
+
+theorem IntegrableOn.mulContinuousOnOfSubset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
+    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
+    IntegrableOn (fun x => g x * g' x) A μ :=
+  by
+  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
+  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg⊢
+  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ :=
+    by
+    filter_upwards [ae_restrict_mem hA]with x hx
+    refine' (norm_mul_le _ _).trans _
+    rw [mul_comm]
+    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
+  exact
+    mem_ℒp.of_le_mul hg (hg.ae_strongly_measurable.mul <| (hg'.mono hAK).AeStronglyMeasurable hA)
+      this
+#align measure_theory.integrable_on.mul_continuous_on_of_subset MeasureTheory.IntegrableOn.mulContinuousOnOfSubset
+
+theorem IntegrableOn.mulContinuousOn [T2Space X] (hg : IntegrableOn g K μ) (hg' : ContinuousOn g' K)
+    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
+  hg.mulContinuousOnOfSubset hg' hK.MeasurableSet hK (Subset.refl _)
+#align measure_theory.integrable_on.mul_continuous_on MeasureTheory.IntegrableOn.mulContinuousOn
+
+theorem IntegrableOn.continuousOnMulOfSubset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
+    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
+    IntegrableOn (fun x => g x * g' x) A μ :=
+  by
+  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
+  rw [integrable_on, ← mem_ℒp_one_iff_integrable] at hg'⊢
+  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ :=
+    by
+    filter_upwards [ae_restrict_mem hA]with x hx
+    refine' (norm_mul_le _ _).trans _
+    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
+  exact
+    mem_ℒp.of_le_mul hg' (((hg.mono hAK).AeStronglyMeasurable hA).mul hg'.ae_strongly_measurable)
+      this
+#align measure_theory.integrable_on.continuous_on_mul_of_subset MeasureTheory.IntegrableOn.continuousOnMulOfSubset
+
+theorem IntegrableOn.continuousOnMul [T2Space X] (hg : ContinuousOn g K) (hg' : IntegrableOn g' K μ)
+    (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ :=
+  hg'.continuousOnMulOfSubset hg hK hK.MeasurableSet Subset.rfl
+#align measure_theory.integrable_on.continuous_on_mul MeasureTheory.IntegrableOn.continuousOnMul
+
+end Mul
+
+section Smul
+
+variable {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 E]
+
+theorem IntegrableOn.continuousOnSmul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
+    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
+    IntegrableOn (fun x => f x • g x) K μ :=
+  by
+  rw [integrable_on, ← integrable_norm_iff]
+  · simp_rw [norm_smul]
+    refine' integrable_on.continuous_on_mul _ hg.norm hK
+    exact continuous_norm.comp_continuous_on hf
+  · exact (hf.ae_strongly_measurable hK.measurable_set).smul hg.1
+#align measure_theory.integrable_on.continuous_on_smul MeasureTheory.IntegrableOn.continuousOnSmul
+
+theorem IntegrableOn.smulContinuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
+    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
+    IntegrableOn (fun x => f x • g x) K μ :=
+  by
+  rw [integrable_on, ← integrable_norm_iff]
+  · simp_rw [norm_smul]
+    refine' integrable_on.mul_continuous_on hf.norm _ hK
+    exact continuous_norm.comp_continuous_on hg
+  · exact hf.1.smul (hg.ae_strongly_measurable hK.measurable_set)
+#align measure_theory.integrable_on.smul_continuous_on MeasureTheory.IntegrableOn.smulContinuousOn
+
+end Smul
+
+namespace LocallyIntegrableOn
+
+theorem continuousOnMul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
+    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
+    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ :=
+  by
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOnMul (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOnMul
+
+theorem mulContinuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
+    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
+    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ :=
+  by
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mulContinuousOn (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mulContinuousOn
+
+theorem continuousOnSmul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+    [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
+    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
+    LocallyIntegrableOn (fun x => g x • f x) s μ :=
+  by
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOnSmul (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOnSmul
+
+theorem smulContinuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+    [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
+    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
+    LocallyIntegrableOn (fun x => f x • g x) s μ :=
+  by
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smulContinuousOn (hg.mono hk_sub) hk_c
+#align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smulContinuousOn
+
+end LocallyIntegrableOn
+
+end MeasureTheory
+

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

@@ -260,7 +260,7 @@ theorem Continuous.integrableOfHasCompactSupport (hf : Continuous f) (hcf : HasC
 
 end borel
 
-open Ennreal
+open ENNReal
 
 section Monotone
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Shortcuts for linearly ordered domains and/or continuous functions. As an example, I golf the existing integrable_of_isBigO_exp_neg.

Another example usage: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/blob/1909a40253607bd2df18a738fc504fe81b132974/PrimeNumberTheoremAnd/PerronFormula.lean#L414-L436

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -389,6 +389,58 @@ theorem integrable_iff_integrableAtFilter_cocompact :
   rewrite [← integrableOn_univ, ← compl_union_self s, integrableOn_union]
   exact ⟨(hloc.integrableOn_isCompact htc).mono ht le_rfl, hs⟩
 
+theorem integrable_iff_integrableAtFilter_atBot_atTop [LinearOrder X] [CompactIccSpace X] :
+    Integrable f μ ↔
+    (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ LocallyIntegrable f μ := by
+  constructor
+  · exact fun hf ↦ ⟨⟨hf.integrableAtFilter _, hf.integrableAtFilter _⟩, hf.locallyIntegrable⟩
+  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
+    exact (IntegrableAtFilter.sup_iff.mpr h.1).filter_mono cocompact_le_atBot_atTop
+
+theorem integrable_iff_integrableAtFilter_atBot [LinearOrder X] [OrderTop X] [CompactIccSpace X] :
+    Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrable f μ := by
+  constructor
+  · exact fun hf ↦ ⟨hf.integrableAtFilter _, hf.locallyIntegrable⟩
+  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
+    exact h.1.filter_mono cocompact_le_atBot
+
+theorem integrable_iff_integrableAtFilter_atTop [LinearOrder X] [OrderBot X] [CompactIccSpace X] :
+    Integrable f μ ↔ IntegrableAtFilter f atTop μ ∧ LocallyIntegrable f μ :=
+  integrable_iff_integrableAtFilter_atBot (X := Xᵒᵈ)
+
+variable {a : X}
+
+theorem integrableOn_Iic_iff_integrableAtFilter_atBot [LinearOrder X] [CompactIccSpace X] :
+    IntegrableOn f (Iic a) μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn f (Iic a) μ := by
+  refine ⟨fun h ↦ ⟨⟨Iic a, Iic_mem_atBot a, h⟩, h.locallyIntegrableOn⟩, fun ⟨⟨s, hsl, hs⟩, h⟩ ↦ ?_⟩
+  haveI : Nonempty X := Nonempty.intro a
+  obtain ⟨a', ha'⟩ := mem_atBot_sets.mp hsl
+  refine (integrableOn_union.mpr ⟨hs.mono ha' le_rfl, ?_⟩).mono Iic_subset_Iic_union_Icc le_rfl
+  exact h.integrableOn_compact_subset Icc_subset_Iic_self isCompact_Icc
+
+theorem integrableOn_Ici_iff_integrableAtFilter_atTop [LinearOrder X] [CompactIccSpace X] :
+    IntegrableOn f (Ici a) μ ↔ IntegrableAtFilter f atTop μ ∧ LocallyIntegrableOn f (Ici a) μ :=
+  integrableOn_Iic_iff_integrableAtFilter_atBot (X := Xᵒᵈ)
+
+theorem integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin
+    [LinearOrder X] [CompactIccSpace X] [NoMinOrder X] [OrderTopology X] :
+    IntegrableOn f (Iio a) μ ↔ IntegrableAtFilter f atBot μ ∧
+    IntegrableAtFilter f (𝓝[<] a) μ ∧ LocallyIntegrableOn f (Iio a) μ := by
+  constructor
+  · intro h
+    exact ⟨⟨Iio a, Iio_mem_atBot a, h⟩, ⟨Iio a, self_mem_nhdsWithin, h⟩, h.locallyIntegrableOn⟩
+  · intro ⟨hbot, ⟨s, hsl, hs⟩, hlocal⟩
+    obtain ⟨s', ⟨hs'_mono, hs'⟩⟩ := mem_nhdsWithin_Iio_iff_exists_Ioo_subset.mp hsl
+    refine (integrableOn_union.mpr ⟨?_, hs.mono hs' le_rfl⟩).mono Iio_subset_Iic_union_Ioo le_rfl
+    exact integrableOn_Iic_iff_integrableAtFilter_atBot.mpr
+      ⟨hbot, hlocal.mono_set (Iic_subset_Iio.mpr hs'_mono)⟩
+
+theorem integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin
+    [LinearOrder X] [CompactIccSpace X] [NoMaxOrder X] [OrderTopology X] :
+    IntegrableOn f (Ioi a) μ ↔ IntegrableAtFilter f atTop μ ∧
+    IntegrableAtFilter f (𝓝[>] a) μ ∧ LocallyIntegrableOn f (Ioi a) μ :=
+  integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin (X := Xᵒᵈ)
+
 end MeasureTheory
 
 open MeasureTheory

This is presumably not exhaustive, but covers about a hundred instances.

Style opinions (e.g., why a particular change is great/not a good idea) are very welcome; I'm still forming my own.

@@ -489,7 +489,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
   have A : IntegrableOn (fun _ => C) s μ := by
     simp only [hs.lt_top, integrableOn_const, or_true_iff]
-  refine'
+  exact
     Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -29,9 +29,7 @@ open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
 open scoped Topology Interval ENNReal BigOperators
 
 variable {X Y E F R : Type*} [MeasurableSpace X] [TopologicalSpace X]
-
 variable [MeasurableSpace Y] [TopologicalSpace Y]
-
 variable [NormedAddCommGroup E] [NormedAddCommGroup F] {f g : X → E} {μ : Measure X} {s : Set X}
 
 namespace MeasureTheory
@@ -398,7 +396,6 @@ open MeasureTheory
 section borel
 
 variable [OpensMeasurableSpace X]
-
 variable {K : Set X} {a b : X}
 
 /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -521,8 +521,8 @@ theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone
     LocallyIntegrable f μ := by
   intro x
   rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
-  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
-  exact exists_Icc_mem_subset_of_mem_nhds hU
+  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U :=
+    exists_Icc_mem_subset_of_mem_nhds hU
   have ab : a ≤ b := xab.1.trans xab.2
   refine' ⟨Icc a b, hab, _⟩
   exact

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -110,7 +110,7 @@ theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
   rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
   let T' : Set (Set X) := insert ∅ T
   have T'_count : T'.Countable := Countable.insert ∅ T_count
-  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
+  have T'_ne : T'.Nonempty := by simp only [T', insert_nonempty]
   rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
   refine' ⟨u, _, _, _⟩
   · intro n

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -89,7 +89,7 @@ theorem LocallyIntegrableOn.exists_countable_integrableOn [SecondCountableTopolo
     rintro ⟨x, hx⟩
     rcases hf x hx with ⟨t, ht, h't⟩
     rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
-    refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
+    exact ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
   choose u u_open xu hu using this
   obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s ⊆ ⋃ i ∈ T, u i := by
     have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)

part 1: corollaries such as for atTop ℝ yet to come

Co-authored-by: L Lllvvuu <git@llllvvuu.dev> Co-authored-by: L <git@llllvvuu.dev>

@@ -382,6 +382,15 @@ theorem LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
     exact hf.integrableOn_isCompact hK
   · exact hg.memℒp_top_of_hasCompactSupport h'g μ
 
+open Filter
+
+theorem integrable_iff_integrableAtFilter_cocompact :
+    Integrable f μ ↔ (IntegrableAtFilter f (cocompact X) μ ∧ LocallyIntegrable f μ) := by
+  refine ⟨fun hf ↦ ⟨hf.integrableAtFilter _, hf.locallyIntegrable⟩, fun ⟨⟨s, hsc, hs⟩, hloc⟩ ↦ ?_⟩
+  obtain ⟨t, htc, ht⟩ := mem_cocompact'.mp hsc
+  rewrite [← integrableOn_univ, ← compl_union_self s, integrableOn_union]
+  exact ⟨(hloc.integrableOn_isCompact htc).mono ht le_rfl, hs⟩
+
 end MeasureTheory
 
 open MeasureTheory

A stronger version of #8800, the differences are:

• assume either IsSigmaCompact U or SigmaCompactSpace M;

• only need test functions satisfying tsupport g ⊆ U rather than support g ⊆ U;

• requires LocallyIntegrableOn U rather than LocallyIntegrable on the whole space.

Also fills in some missing APIs around the manifold and measure theory libraries.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -28,11 +28,11 @@ open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
 
 open scoped Topology Interval ENNReal BigOperators
 
-variable {X Y E R : Type*} [MeasurableSpace X] [TopologicalSpace X]
+variable {X Y E F R : Type*} [MeasurableSpace X] [TopologicalSpace X]
 
 variable [MeasurableSpace Y] [TopologicalSpace Y]
 
-variable [NormedAddCommGroup E] {f g : X → E} {μ : Measure X} {s : Set X}
+variable [NormedAddCommGroup E] [NormedAddCommGroup F] {f g : X → E} {μ : Measure X} {s : Set X}
 
 namespace MeasureTheory
 
@@ -45,10 +45,10 @@ def LocallyIntegrableOn (f : X → E) (s : Set X) (μ : Measure X := by volume_t
   ∀ x : X, x ∈ s → IntegrableAtFilter f (𝓝[s] x) μ
 #align measure_theory.locally_integrable_on MeasureTheory.LocallyIntegrableOn
 
-theorem LocallyIntegrableOn.mono (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X}
+theorem LocallyIntegrableOn.mono_set (hf : LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) : LocallyIntegrableOn f t μ := fun x hx =>
   (hf x <| hst hx).filter_mono (nhdsWithin_mono x hst)
-#align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono
+#align measure_theory.locally_integrable_on.mono MeasureTheory.LocallyIntegrableOn.mono_set
 
 theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
     LocallyIntegrableOn (fun x => ‖f x‖) s μ := fun t ht =>
@@ -56,6 +56,13 @@ theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
   ⟨U, hU_nhd, hU_int.norm⟩
 #align measure_theory.locally_integrable_on.norm MeasureTheory.LocallyIntegrableOn.norm
 
+theorem LocallyIntegrableOn.mono (hf : LocallyIntegrableOn f s μ) {g : X → F}
+    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
+    LocallyIntegrableOn g s μ := by
+  intro x hx
+  rcases hf x hx with ⟨t, t_mem, ht⟩
+  exact ⟨t, t_mem, Integrable.mono ht hg.restrict (ae_restrict_of_ae h)⟩
+
 theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyIntegrableOn f s μ :=
   fun _ _ => ⟨s, self_mem_nhdsWithin, hf⟩
 #align measure_theory.integrable_on.locally_integrable_on MeasureTheory.IntegrableOn.locallyIntegrableOn
@@ -69,7 +76,7 @@ theorem LocallyIntegrableOn.integrableOn_isCompact (hf : LocallyIntegrableOn f s
 
 theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableOn f s μ) {t : Set X}
     (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t μ :=
-  (hf.mono hst).integrableOn_isCompact ht
+  (hf.mono_set hst).integrableOn_isCompact ht
 #align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
 
 /-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
@@ -171,6 +178,11 @@ def LocallyIntegrable (f : X → E) (μ : Measure X := by volume_tac) : Prop :=
   ∀ x : X, IntegrableAtFilter f (𝓝 x) μ
 #align measure_theory.locally_integrable MeasureTheory.LocallyIntegrable
 
+theorem locallyIntegrable_comap (hs : MeasurableSet s) :
+    LocallyIntegrable (fun x : s ↦ f x) (μ.comap Subtype.val) ↔ LocallyIntegrableOn f s μ := by
+  simp_rw [LocallyIntegrableOn, Subtype.forall', ← map_nhds_subtype_val]
+  exact forall_congr' fun _ ↦ (MeasurableEmbedding.subtype_coe hs).integrableAtFilter_iff_comap.symm
+
 theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ LocallyIntegrable f μ := by
   simp only [LocallyIntegrableOn, nhdsWithin_univ, mem_univ, true_imp_iff]; rfl
 #align measure_theory.locally_integrable_on_univ MeasureTheory.locallyIntegrableOn_univ
@@ -183,6 +195,12 @@ theorem Integrable.locallyIntegrable (hf : Integrable f μ) : LocallyIntegrable
   hf.integrableAtFilter _
 #align measure_theory.integrable.locally_integrable MeasureTheory.Integrable.locallyIntegrable
 
+theorem LocallyIntegrable.mono (hf : LocallyIntegrable f μ) {g : X → F}
+    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
+    LocallyIntegrable g μ := by
+  rw [← locallyIntegrableOn_univ] at hf ⊢
+  exact hf.mono hg h
+
 /-- If `f` is locally integrable with respect to `μ.restrict s`, it is locally integrable on `s`.
 (See `locallyIntegrableOn_iff_locallyIntegrable_restrict` for an iff statement when `s` is
 closed.) -/

Also weaken some T2 space assumptions in some integration lemmas.

Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com>

@@ -370,61 +370,70 @@ open MeasureTheory
 
 section borel
 
-variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
+variable [OpensMeasurableSpace X]
 
 variable {K : Set X} {a b : X}
 
 /-- A continuous function `f` is locally integrable with respect to any locally finite measure. -/
-theorem Continuous.locallyIntegrable [SecondCountableTopologyEither X E] (hf : Continuous f) :
-    LocallyIntegrable f μ :=
+theorem Continuous.locallyIntegrable [IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E]
+    (hf : Continuous f) : LocallyIntegrable f μ :=
   hf.integrableAt_nhds
 #align continuous.locally_integrable Continuous.locallyIntegrable
 
 /-- A function `f` continuous on a set `K` is locally integrable on this set with respect
 to any locally finite measure. -/
-theorem ContinuousOn.locallyIntegrableOn [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
+theorem ContinuousOn.locallyIntegrableOn [IsLocallyFiniteMeasure μ]
+    [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
     (hK : MeasurableSet K) : LocallyIntegrableOn f K μ := fun _x hx =>
   hf.integrableAt_nhdsWithin hK hx
 #align continuous_on.locally_integrable_on ContinuousOn.locallyIntegrableOn
 
-variable [MetrizableSpace X]
+variable [IsFiniteMeasureOnCompacts μ]
 
 /-- A function `f` continuous on a compact set `K` is integrable on this set with respect to any
 locally finite measure. -/
-theorem ContinuousOn.integrableOn_compact (hK : IsCompact K) (hf : ContinuousOn f K) :
+theorem ContinuousOn.integrableOn_compact'
+    (hK : IsCompact K) (h'K : MeasurableSet K) (hf : ContinuousOn f K) :
     IntegrableOn f K μ := by
-  letI := metrizableSpaceMetric X
-  refine' LocallyIntegrableOn.integrableOn_isCompact (fun x hx => _) hK
-  exact hf.integrableAt_nhdsWithin_of_isSeparable hK.measurableSet hK.isSeparable hx
+  refine ⟨ContinuousOn.aestronglyMeasurable_of_isCompact hf hK h'K, ?_⟩
+  have : Fact (μ K < ∞) := ⟨hK.measure_lt_top⟩
+  obtain ⟨C, hC⟩ : ∃ C, ∀ x ∈ f '' K, ‖x‖ ≤ C :=
+    IsBounded.exists_norm_le (hK.image_of_continuousOn hf).isBounded
+  apply hasFiniteIntegral_of_bounded (C := C)
+  filter_upwards [ae_restrict_mem h'K] with x hx using hC _ (mem_image_of_mem f hx)
+
+theorem ContinuousOn.integrableOn_compact [T2Space X]
+    (hK : IsCompact K) (hf : ContinuousOn f K) : IntegrableOn f K μ :=
+  hf.integrableOn_compact' hK hK.measurableSet
 #align continuous_on.integrable_on_compact ContinuousOn.integrableOn_compact
 
-theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X]
+theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
     (hf : ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) μ :=
   hf.integrableOn_compact isCompact_Icc
 #align continuous_on.integrable_on_Icc ContinuousOn.integrableOn_Icc
 
-theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
-    IntegrableOn f (Icc a b) μ :=
+theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
+    (hf : Continuous f) : IntegrableOn f (Icc a b) μ :=
   hf.continuousOn.integrableOn_Icc
 #align continuous.integrable_on_Icc Continuous.integrableOn_Icc
 
-theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] (hf : Continuous f) :
-    IntegrableOn f (Ioc a b) μ :=
+theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] [T2Space X]
+    (hf : Continuous f) : IntegrableOn f (Ioc a b) μ :=
   hf.integrableOn_Icc.mono_set Ioc_subset_Icc_self
 #align continuous.integrable_on_Ioc Continuous.integrableOn_Ioc
 
-theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X]
+theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
     (hf : ContinuousOn f [[a, b]]) : IntegrableOn f [[a, b]] μ :=
   hf.integrableOn_Icc
 #align continuous_on.integrable_on_uIcc ContinuousOn.integrableOn_uIcc
 
-theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
-    IntegrableOn f [[a, b]] μ :=
+theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
+    (hf : Continuous f) : IntegrableOn f [[a, b]] μ :=
   hf.integrableOn_Icc
 #align continuous.integrable_on_uIcc Continuous.integrableOn_uIcc
 
-theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] (hf : Continuous f) :
-    IntegrableOn f (Ι a b) μ :=
+theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] [T2Space X]
+    (hf : Continuous f) : IntegrableOn f (Ι a b) μ :=
   hf.integrableOn_Ioc
 #align continuous.integrable_on_uIoc Continuous.integrableOn_uIoc
 
@@ -432,7 +441,7 @@ theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] (hf : C
 theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
     Integrable f μ :=
   (integrableOn_iff_integrable_of_support_subset (subset_tsupport f)).mp <|
-    hf.continuousOn.integrableOn_compact hcf
+    hf.continuousOn.integrableOn_compact' hcf (isClosed_tsupport _).measurableSet
 #align continuous.integrable_of_has_compact_support Continuous.integrable_of_hasCompactSupport
 
 end borel

mathport was forgetting a space in filter_upwards [...]with instead of filter_upwards [...] with.

@@ -515,7 +515,7 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
   rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
   rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
-    filter_upwards [ae_restrict_mem hA]with x hx
+    filter_upwards [ae_restrict_mem hA] with x hx
     refine' (norm_mul_le _ _).trans _
     rw [mul_comm]
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
@@ -534,7 +534,7 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
   rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
   rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
-    filter_upwards [ae_restrict_mem hA]with x hx
+    filter_upwards [ae_restrict_mem hA] with x hx
     refine' (norm_mul_le _ _).trans _
     apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
   exact

Rename union_eq_left_iff_subset to union_eq_left to match sup_eq_left. Similarly for the right and inter versions.

@@ -126,7 +126,7 @@ theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
 theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
     (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) := by
   rcases hf.exists_nat_integrableOn with ⟨u, -, su, hu⟩
-  have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right_iff_subset.mpr su).symm
+  have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right.mpr su).symm
   rw [this, aestronglyMeasurable_iUnion_iff]
   exact fun i : ℕ => (hu i).aestronglyMeasurable
 #align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable

As discussed on Zulip.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -144,7 +144,7 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
       let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
       ⟨_, inter_mem_nhdsWithin s h2K,
         hf _ (inter_subset_left _ _)
-          (isCompact_of_isClosed_subset hK (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
+          (hK.of_isClosed_subset (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
   | inr hs =>
     obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
     refine' ⟨K, _, hf K h3K hK⟩

Use Bornology.IsBounded instead.

@@ -24,7 +24,7 @@ on compact sets.
   integrable on `s`.
 -/
 
-open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace
+open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
 
 open scoped Topology Interval ENNReal BigOperators
 
@@ -452,8 +452,8 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   · exact integrableOn_empty
   have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
   have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
-  have : Metric.Bounded (f '' s) := Metric.bounded_of_bddAbove_of_bddBelow habove hbelow
-  rcases bounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
+  have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
+  rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
   have A : IntegrableOn (fun _ => C) s μ := by
     simp only [hs.lt_top, integrableOn_const, or_true_iff]
   refine'

Measure theory prerequisites for Rademacher theorem in #7003.

@@ -24,18 +24,15 @@ on compact sets.
   integrable on `s`.
 -/
 
-set_option autoImplicit true
-
-
 open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace
 
-open scoped Topology Interval
+open scoped Topology Interval ENNReal BigOperators
 
 variable {X Y E R : Type*} [MeasurableSpace X] [TopologicalSpace X]
 
 variable [MeasurableSpace Y] [TopologicalSpace Y]
 
-variable [NormedAddCommGroup E] {f : X → E} {μ : Measure X} {s : Set X}
+variable [NormedAddCommGroup E] {f g : X → E} {μ : Measure X} {s : Set X}
 
 namespace MeasureTheory
 
@@ -260,12 +257,18 @@ theorem LocallyIntegrable.exists_nat_integrableOn [SecondCountableTopology X]
   refine' ⟨u, u_open, eq_univ_of_univ_subset u_union, fun n ↦ _⟩
   simpa only [inter_univ] using hu n
 
-theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
-    LocallyIntegrable (fun _ => c) μ := by
+theorem Memℒp.locallyIntegrable [IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞}
+    (hf : Memℒp f p μ) (hp : 1 ≤ p) : LocallyIntegrable f μ := by
   intro x
   rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
+  have : Fact (μ U < ⊤) := ⟨h'U⟩
   refine' ⟨U, hU, _⟩
-  simp only [h'U, integrableOn_const, or_true_iff]
+  rw [IntegrableOn, ← memℒp_one_iff_integrable]
+  apply (hf.restrict U).memℒp_of_exponent_le hp
+
+theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
+    LocallyIntegrable (fun _ => c) μ :=
+  (memℒp_top_const c).locallyIntegrable le_top
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
 
 theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
@@ -273,6 +276,12 @@ theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
   (locallyIntegrable_const c).locallyIntegrableOn s
 #align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
 
+theorem locallyIntegrable_zero : LocallyIntegrable (fun _ ↦ (0 : E)) μ :=
+  (integrable_zero X E μ).locallyIntegrable
+
+theorem locallyIntegrableOn_zero : LocallyIntegrableOn (fun _ ↦ (0 : E)) s μ :=
+  locallyIntegrable_zero.locallyIntegrableOn s
+
 theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
     (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ := by
   intro x
@@ -304,6 +313,19 @@ protected theorem LocallyIntegrable.sub (hf : LocallyIntegrable f μ) (hg : Loca
 protected theorem LocallyIntegrable.neg (hf : LocallyIntegrable f μ) :
     LocallyIntegrable (-f) μ := fun x ↦ (hf x).neg
 
+protected theorem LocallyIntegrable.smul {𝕜 : Type*} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E]
+    [BoundedSMul 𝕜 E] (hf : LocallyIntegrable f μ) (c : 𝕜) :
+    LocallyIntegrable (c • f) μ := fun x ↦ (hf x).smul c
+
+theorem locallyIntegrable_finset_sum' {ι} (s : Finset ι) {f : ι → X → E}
+    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (∑ i in s, f i) μ :=
+  Finset.sum_induction f (fun g => LocallyIntegrable g μ) (fun _ _ => LocallyIntegrable.add)
+    locallyIntegrable_zero hf
+
+theorem locallyIntegrable_finset_sum {ι} (s : Finset ι) {f : ι → X → E}
+    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (fun a ↦ ∑ i in s, f i a) μ := by
+  simpa only [← Finset.sum_apply] using locallyIntegrable_finset_sum' s hf
+
 /-- If `f` is locally integrable and `g` is continuous with compact support,
 then `g • f` is integrable. -/
 theorem LocallyIntegrable.integrable_smul_left_of_hasCompactSupport

@@ -526,7 +526,7 @@ theorem IntegrableOn.continuousOn_mul [T2Space X] (hg : ContinuousOn g K)
 
 end Mul
 
-section Smul
+section SMul
 
 variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
 
@@ -550,7 +550,7 @@ theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEithe
   · exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)
 #align measure_theory.integrable_on.smul_continuous_on MeasureTheory.IntegrableOn.smul_continuousOn
 
-end Smul
+end SMul
 
 namespace LocallyIntegrableOn
 

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -24,6 +24,8 @@ on compact sets.
   integrable on `s`.
 -/
 
+set_option autoImplicit true
+
 
 open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace
 

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -29,7 +29,7 @@ open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace
 
 open scoped Topology Interval
 
-variable {X Y E R : Type _} [MeasurableSpace X] [TopologicalSpace X]
+variable {X Y E R : Type*} [MeasurableSpace X] [TopologicalSpace X]
 
 variable [MeasurableSpace Y] [TopologicalSpace Y]
 
@@ -526,7 +526,7 @@ end Mul
 
 section Smul
 
-variable {𝕜 : Type _} [NormedField 𝕜] [NormedSpace 𝕜 E]
+variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
 
 theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
     (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
@@ -566,7 +566,7 @@ theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mul_continuousOn
 
-theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
     [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => g x • f x) s μ := by
@@ -574,7 +574,7 @@ theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOn_smul
 
-theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [NormedField 𝕜]
+theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
     [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => f x • g x) s μ := by

@@ -302,6 +302,44 @@ protected theorem LocallyIntegrable.sub (hf : LocallyIntegrable f μ) (hg : Loca
 protected theorem LocallyIntegrable.neg (hf : LocallyIntegrable f μ) :
     LocallyIntegrable (-f) μ := fun x ↦ (hf x).neg
 
+/-- If `f` is locally integrable and `g` is continuous with compact support,
+then `g • f` is integrable. -/
+theorem LocallyIntegrable.integrable_smul_left_of_hasCompactSupport
+    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X]
+    (hf : LocallyIntegrable f μ) {g : X → ℝ} (hg : Continuous g) (h'g : HasCompactSupport g) :
+    Integrable (fun x ↦ g x • f x) μ := by
+  let K := tsupport g
+  have hK : IsCompact K := h'g
+  have : K.indicator (fun x ↦ g x • f x) = (fun x ↦ g x • f x) := by
+    apply indicator_eq_self.2
+    apply support_subset_iff'.2
+    intros x hx
+    simp [image_eq_zero_of_nmem_tsupport hx]
+  rw [← this, indicator_smul]
+  apply Integrable.smul_of_top_right
+  · rw [integrable_indicator_iff hK.measurableSet]
+    exact hf.integrableOn_isCompact hK
+  · exact hg.memℒp_top_of_hasCompactSupport h'g μ
+
+/-- If `f` is locally integrable and `g` is continuous with compact support,
+then `f • g` is integrable. -/
+theorem LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
+    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] {f : X → ℝ} (hf : LocallyIntegrable f μ)
+    {g : X → E} (hg : Continuous g) (h'g : HasCompactSupport g) :
+    Integrable (fun x ↦ f x • g x) μ := by
+  let K := tsupport g
+  have hK : IsCompact K := h'g
+  have : K.indicator (fun x ↦ f x • g x) = (fun x ↦ f x • g x) := by
+    apply indicator_eq_self.2
+    apply support_subset_iff'.2
+    intros x hx
+    simp [image_eq_zero_of_nmem_tsupport hx]
+  rw [← this, indicator_smul_left]
+  apply Integrable.smul_of_top_left
+  · rw [integrable_indicator_iff hK.measurableSet]
+    exact hf.integrableOn_isCompact hK
+  · exact hg.memℒp_top_of_hasCompactSupport h'g μ
+
 end MeasureTheory
 
 open MeasureTheory

We show that, if a locally integrable function has zero integral on all compact sets, then it vanishes almost everywhere.

@@ -73,24 +73,63 @@ theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableO
   (hf.mono hst).integrableOn_isCompact ht
 #align measure_theory.locally_integrable_on.integrable_on_compact_subset MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
 
-theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
-    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) := by
+/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
+then there exist countably many open sets `u` covering `s` such that `f` is integrable on each
+set `u ∩ s`. -/
+theorem LocallyIntegrableOn.exists_countable_integrableOn [SecondCountableTopology X]
+    (hf : LocallyIntegrableOn f s μ) : ∃ T : Set (Set X), T.Countable ∧
+    (∀ u ∈ T, IsOpen u) ∧ (s ⊆ ⋃ u ∈ T, u) ∧ (∀ u ∈ T, IntegrableOn f (u ∩ s) μ) := by
   have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ := by
     rintro ⟨x, hx⟩
     rcases hf x hx with ⟨t, ht, h't⟩
     rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
     refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
   choose u u_open xu hu using this
-  obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s = ⋃ i : T, u i ∩ s := by
+  obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s ⊆ ⋃ i ∈ T, u i := by
     have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
     obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open
-    refine' ⟨T, hT_count, _⟩
-    rw [← hT_un, biUnion_eq_iUnion] at this
-    rw [← iUnion_inter, eq_comm, inter_eq_right_iff_subset]
-    exact this
-  have : Countable T := countable_coe_iff.mpr T_count
-  rw [hT, aestronglyMeasurable_iUnion_iff]
-  exact fun i : T => (hu i).aestronglyMeasurable
+    exact ⟨T, hT_count, by rwa [hT_un]⟩
+  refine' ⟨u '' T, T_count.image _, _, by rwa [biUnion_image], _⟩
+  · rintro v ⟨w, -, rfl⟩
+    exact u_open _
+  · rintro v ⟨w, -, rfl⟩
+    exact hu _
+
+/-- If a function `f` is locally integrable on a set `s` in a second countable topological space,
+then there exists a sequence of open sets `u n` covering `s` such that `f` is integrable on each
+set `u n ∩ s`. -/
+theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
+    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
+    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ) := by
+  rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
+  let T' : Set (Set X) := insert ∅ T
+  have T'_count : T'.Countable := Countable.insert ∅ T_count
+  have T'_ne : T'.Nonempty := by simp only [insert_nonempty]
+  rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩
+  refine' ⟨u, _, _, _⟩
+  · intro n
+    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
+    rcases mem_insert_iff.1 this with h|h
+    · rw [h]
+      exact isOpen_empty
+    · exact T_open _ h
+  · intro x hx
+    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
+    have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
+    obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this
+    exact mem_iUnion_of_mem _ h'v
+  · intro n
+    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
+    rcases mem_insert_iff.1 this with h|h
+    · simp only [h, empty_inter, integrableOn_empty]
+    · exact hT _ h
+
+theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
+    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) := by
+  rcases hf.exists_nat_integrableOn with ⟨u, -, su, hu⟩
+  have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right_iff_subset.mpr su).symm
+  rw [this, aestronglyMeasurable_iUnion_iff]
+  exact fun i : ℕ => (hu i).aestronglyMeasurable
 #align measure_theory.locally_integrable_on.ae_strongly_measurable MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable
 
 /-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
@@ -113,6 +152,17 @@ theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClos
     simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
 #align measure_theory.locally_integrable_on_iff MeasureTheory.locallyIntegrableOn_iff
 
+protected theorem LocallyIntegrableOn.add
+    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
+    LocallyIntegrableOn (f + g) s μ := fun x hx ↦ (hf x hx).add (hg x hx)
+
+protected theorem LocallyIntegrableOn.sub
+    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
+    LocallyIntegrableOn (f - g) s μ := fun x hx ↦ (hf x hx).sub (hg x hx)
+
+protected theorem LocallyIntegrableOn.neg (hf : LocallyIntegrableOn f s μ) :
+    LocallyIntegrableOn (-f) s μ := fun x hx ↦ (hf x hx).neg
+
 end LocallyIntegrableOn
 
 /-- A function `f : X → E` is *locally integrable* if it is integrable on a neighborhood of every
@@ -199,6 +249,15 @@ theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
   simpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aestronglyMeasurable
 
+/-- If a function is locally integrable in a second countable topological space,
+then there exists a sequence of open sets covering the space on which it is integrable. -/
+theorem LocallyIntegrable.exists_nat_integrableOn [SecondCountableTopology X]
+    (hf : LocallyIntegrable f μ) : ∃ u : ℕ → Set X,
+    (∀ n, IsOpen (u n)) ∧ ((⋃ n, u n) = univ) ∧ (∀ n, IntegrableOn f (u n) μ) := by
+  rcases (hf.locallyIntegrableOn univ).exists_nat_integrableOn with ⟨u, u_open, u_union, hu⟩
+  refine' ⟨u, u_open, eq_univ_of_univ_subset u_union, fun n ↦ _⟩
+  simpa only [inter_univ] using hu n
+
 theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun _ => c) μ := by
   intro x
@@ -234,6 +293,15 @@ theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X 
     simp only [mem_preimage, Homeomorph.symm_apply_apply]
 #align measure_theory.locally_integrable_map_homeomorph MeasureTheory.locallyIntegrable_map_homeomorph
 
+protected theorem LocallyIntegrable.add (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
+    LocallyIntegrable (f + g) μ := fun x ↦ (hf x).add (hg x)
+
+protected theorem LocallyIntegrable.sub (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
+    LocallyIntegrable (f - g) μ := fun x ↦ (hf x).sub (hg x)
+
+protected theorem LocallyIntegrable.neg (hf : LocallyIntegrable f μ) :
+    LocallyIntegrable (-f) μ := fun x ↦ (hf x).neg
+
 end MeasureTheory
 
 open MeasureTheory

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
-
-! This file was ported from Lean 3 source module measure_theory.function.locally_integrable
-! leanprover-community/mathlib commit 08a4542bec7242a5c60f179e4e49de8c0d677b1b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.MeasureTheory.Integral.IntegrableOn
 
+#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
+
 /-!
 # Locally integrable functions
 

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -386,7 +386,7 @@ theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg'
     (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ := by
   rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
-  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg⊢
+  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
     filter_upwards [ae_restrict_mem hA]with x hx
     refine' (norm_mul_le _ _).trans _
@@ -405,7 +405,7 @@ theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : I
     (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
     IntegrableOn (fun x => g x * g' x) A μ := by
   rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
-  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg'⊢
+  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
   have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
     filter_upwards [ae_restrict_mem hA]with x hx
     refine' (norm_mul_le _ _).trans _
@@ -452,14 +452,14 @@ namespace LocallyIntegrableOn
 theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_mul MeasureTheory.LocallyIntegrableOn.continuousOn_mul
 
 theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
     (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.mul_continuous_on MeasureTheory.LocallyIntegrableOn.mul_continuousOn
 
@@ -467,7 +467,7 @@ theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
     [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => g x • f x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.continuous_on_smul MeasureTheory.LocallyIntegrableOn.continuousOn_smul
 
@@ -475,7 +475,7 @@ theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type _} [N
     [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
     (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => f x • g x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c
 #align measure_theory.locally_integrable_on.smul_continuous_on MeasureTheory.LocallyIntegrableOn.smul_continuousOn
 

I have misported is_foo to Foo because I misunderstood the rule for IsLawfulFoo. This PR recover Is of Foo which is ported from is_foo. This PR also renames some misported theorems.

@@ -202,7 +202,7 @@ theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
   simpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable
 #align measure_theory.locally_integrable.ae_strongly_measurable MeasureTheory.LocallyIntegrable.aestronglyMeasurable
 
-theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrable (fun _ => c) μ := by
   intro x
   rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
@@ -210,7 +210,7 @@ theorem locallyIntegrable_const [LocallyFiniteMeasure μ] (c : E) :
   simp only [h'U, integrableOn_const, or_true_iff]
 #align measure_theory.locally_integrable_const MeasureTheory.locallyIntegrable_const
 
-theorem locallyIntegrableOn_const [LocallyFiniteMeasure μ] (c : E) :
+theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
     LocallyIntegrableOn (fun _ => c) s μ :=
   (locallyIntegrable_const c).locallyIntegrableOn s
 #align measure_theory.locally_integrable_on_const MeasureTheory.locallyIntegrableOn_const
@@ -243,7 +243,7 @@ open MeasureTheory
 
 section borel
 
-variable [OpensMeasurableSpace X] [LocallyFiniteMeasure μ]
+variable [OpensMeasurableSpace X] [IsLocallyFiniteMeasure μ]
 
 variable {K : Set X} {a b : X}
 
@@ -334,7 +334,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
 #align monotone_on.integrable_on_of_measure_ne_top MonotoneOn.integrableOn_of_measure_ne_top
 
-theorem MonotoneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hmono : MonotoneOn f s) : IntegrableOn f s μ := by
   obtain rfl | h := s.eq_empty_or_nonempty
   · exact integrableOn_empty
@@ -349,12 +349,12 @@ theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b
   hanti.dual_right.integrableOn_of_measure_ne_top ha hb hs h's
 #align antitone_on.integrable_on_of_measure_ne_top AntitoneOn.integrableOn_of_measure_ne_top
 
-theorem AntioneOn.integrableOn_isCompact [FiniteMeasureOnCompacts μ] (hs : IsCompact s)
+theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
     (hanti : AntitoneOn f s) : IntegrableOn f s μ :=
   hanti.dual_right.integrableOn_isCompact (E := Eᵒᵈ) hs
 #align antione_on.integrable_on_is_compact AntioneOn.integrableOn_isCompact
 
-theorem Monotone.locallyIntegrable [LocallyFiniteMeasure μ] (hmono : Monotone f) :
+theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
     LocallyIntegrable f μ := by
   intro x
   rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
@@ -367,7 +367,7 @@ theorem Monotone.locallyIntegrable [LocallyFiniteMeasure μ] (hmono : Monotone f
       ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc
 #align monotone.locally_integrable Monotone.locallyIntegrable
 
-theorem Antitone.locallyIntegrable [LocallyFiniteMeasure μ] (hanti : Antitone f) :
+theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
     LocallyIntegrable f μ :=
   hanti.dual_right.locallyIntegrable
 #align antitone.locally_integrable Antitone.locallyIntegrable