probability.kernel.with_density ⟷ Mathlib.Probability.Kernel.WithDensity

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

@@ -206,7 +206,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
     rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
     · refine' this.trans (le_of_eq _)
-      rw [ENNReal.ofReal_coe_nat]
+      rw [ENNReal.ofReal_natCast]
     · norm_cast
       exact zero_le _
   have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 :=

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

@@ -204,7 +204,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     by
     intro a b n hn
     have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
-    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this 
+    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
     · refine' this.trans (le_of_eq _)
       rw [ENNReal.ofReal_coe_nat]
     · norm_cast
@@ -225,7 +225,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
       suffices : ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0
       exact summable_of_ne_finset_zero this
       intro n hn_not_mem
-      rw [Finset.mem_range, not_lt] at hn_not_mem 
+      rw [Finset.mem_range, not_lt] at hn_not_mem
       exact h_zero a b n hn_not_mem
     ext a b : 2
     rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),

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

@@ -64,14 +64,17 @@ noncomputable def withDensity (κ : kernel α β) [IsSFiniteKernel κ] (f : α 
 
 #print ProbabilityTheory.kernel.withDensity_of_not_measurable /-
 theorem withDensity_of_not_measurable (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical
+    (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical exact dif_neg hf
 #align probability_theory.kernel.with_density_of_not_measurable ProbabilityTheory.kernel.withDensity_of_not_measurable
 -/
 
 #print ProbabilityTheory.kernel.withDensity_apply /-
 protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) : withDensity κ f a = (κ a).withDensity (f a) :=
-  by classical
+  by
+  classical
+  rw [with_density, dif_pos hf]
+  rfl
 #align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
 -/
 

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

@@ -64,17 +64,14 @@ noncomputable def withDensity (κ : kernel α β) [IsSFiniteKernel κ] (f : α 
 
 #print ProbabilityTheory.kernel.withDensity_of_not_measurable /-
 theorem withDensity_of_not_measurable (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical exact dif_neg hf
+    (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical
 #align probability_theory.kernel.with_density_of_not_measurable ProbabilityTheory.kernel.withDensity_of_not_measurable
 -/
 
 #print ProbabilityTheory.kernel.withDensity_apply /-
 protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) : withDensity κ f a = (κ a).withDensity (f a) :=
-  by
-  classical
-  rw [with_density, dif_pos hf]
-  rfl
+  by classical
 #align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
 -/
 

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

@@ -3,8 +3,8 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathbin.Probability.Kernel.MeasurableIntegral
-import Mathbin.MeasureTheory.Integral.SetIntegral
+import Probability.Kernel.MeasurableIntegral
+import MeasureTheory.Integral.SetIntegral
 
 #align_import probability.kernel.with_density from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.with_density
-! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Kernel.MeasurableIntegral
 import Mathbin.MeasureTheory.Integral.SetIntegral
 
+#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"8af7091a43227e179939ba132e54e54e9f3b089a"
+
 /-!
 # With Density
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -117,7 +117,7 @@ theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
     (f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f :=
   by
   by_cases hf : Measurable (Function.uncurry f)
-  · ext (a s hs) : 2
+  · ext a s hs : 2
     simp only [kernel.with_density_apply _ hf, coe_fn_add, Pi.add_apply, with_density_add_measure,
       measure.add_apply]
   · simp_rw [with_density_of_not_measurable _ hf]
@@ -147,7 +147,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
   by
   have h_sum_a : ∀ a, Summable fun n => f n a := fun a => pi.summable.mpr fun b => ENNReal.summable
   have h_sum : Summable fun n => f n := pi.summable.mpr h_sum_a
-  ext (a s hs) : 2
+  ext a s hs : 2
   rw [sum_apply' _ a hs, with_density_apply' κ _ a hs]
   swap
   · have : Function.uncurry (∑' n, f n) = ∑' n, Function.uncurry (f n) :=
@@ -230,7 +230,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
       intro n hn_not_mem
       rw [Finset.mem_range, not_lt] at hn_not_mem 
       exact h_zero a b n hn_not_mem
-    ext (a b) : 2
+    ext a b : 2
     rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
       ENNReal.tsum_eq_liminf_sum_nat]
     have h_finset_sum : ∀ n, ∑ i in Finset.range n, fs i a b = min (f a b) n :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.with_density
-! leanprover-community/mathlib commit c0d694db494dd4f9aa57f2714b6e4c82b4ebc113
+! leanprover-community/mathlib commit 8af7091a43227e179939ba132e54e54e9f3b089a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.MeasureTheory.Integral.SetIntegral
 /-!
 # With Density
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 For an s-finite kernel `κ : kernel α β` and a function `f : α → β → ℝ≥0∞` which is finite
 everywhere, we define `with_density κ f` as the kernel `a ↦ (κ a).with_density (f a)`. This is
 an s-finite kernel.

mathlib commit https://github.com/leanprover-community/mathlib/commit/8efcf8022aac8e01df8d302dcebdbc25d6a886c8

@@ -44,6 +44,7 @@ variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace 
 
 variable {κ : kernel α β} {f : α → β → ℝ≥0∞}
 
+#print ProbabilityTheory.kernel.withDensity /-
 /-- Kernel with image `(κ a).with_density (f a)` if `function.uncurry f` is measurable, and
 with image 0 otherwise. If `function.uncurry f` is measurable, it satisfies
 `∫⁻ b, g b ∂(with_density κ f hf a) = ∫⁻ b, f a b * g b ∂(κ a)`. -/
@@ -59,11 +60,15 @@ noncomputable def withDensity (κ : kernel α β) [IsSFiniteKernel κ] (f : α 
         kernel α β))
     fun hf => 0
 #align probability_theory.kernel.with_density ProbabilityTheory.kernel.withDensity
+-/
 
+#print ProbabilityTheory.kernel.withDensity_of_not_measurable /-
 theorem withDensity_of_not_measurable (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : ¬Measurable (Function.uncurry f)) : withDensity κ f = 0 := by classical exact dif_neg hf
 #align probability_theory.kernel.with_density_of_not_measurable ProbabilityTheory.kernel.withDensity_of_not_measurable
+-/
 
+#print ProbabilityTheory.kernel.withDensity_apply /-
 protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) : withDensity κ f a = (κ a).withDensity (f a) :=
   by
@@ -71,13 +76,17 @@ protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
   rw [with_density, dif_pos hf]
   rfl
 #align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
+-/
 
+#print ProbabilityTheory.kernel.withDensity_apply' /-
 theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) {s : Set β} (hs : MeasurableSet s) :
     withDensity κ f a s = ∫⁻ b in s, f a b ∂κ a := by
   rw [kernel.with_density_apply κ hf, with_density_apply _ hs]
 #align probability_theory.kernel.with_density_apply' ProbabilityTheory.kernel.withDensity_apply'
+-/
 
+#print ProbabilityTheory.kernel.lintegral_withDensity /-
 theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
     ∫⁻ b, g b ∂withDensity κ f a = ∫⁻ b, f a b * g b ∂κ a :=
@@ -86,7 +95,9 @@ theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     lintegral_with_density_eq_lintegral_mul _ (Measurable.of_uncurry_left hf) hg]
   simp_rw [Pi.mul_apply]
 #align probability_theory.kernel.lintegral_with_density ProbabilityTheory.kernel.lintegral_withDensity
+-/
 
+#print ProbabilityTheory.kernel.integral_withDensity /-
 theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
     (hg : Measurable (Function.uncurry g)) :
@@ -96,7 +107,9 @@ theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace 
   · exact Measurable.of_uncurry_left hg
   · exact measurable_coe_nnreal_ennreal.comp hg
 #align probability_theory.kernel.integral_with_density ProbabilityTheory.kernel.integral_withDensity
+-/
 
+#print ProbabilityTheory.kernel.withDensity_add_left /-
 theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
     (f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f :=
   by
@@ -107,7 +120,9 @@ theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFin
   · simp_rw [with_density_of_not_measurable _ hf]
     rw [zero_add]
 #align probability_theory.kernel.with_density_add_left ProbabilityTheory.kernel.withDensity_add_left
+-/
 
+#print ProbabilityTheory.kernel.withDensity_kernel_sum /-
 theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
     (f : α → β → ℝ≥0∞) :
     @withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
@@ -120,7 +135,9 @@ theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ :
   · simp_rw [with_density_of_not_measurable _ hf]
     exact sum_zero.symm
 #align probability_theory.kernel.with_density_kernel_sum ProbabilityTheory.kernel.withDensity_kernel_sum
+-/
 
+#print ProbabilityTheory.kernel.withDensity_tsum /-
 theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ] {f : ι → α → β → ℝ≥0∞}
     (hf : ∀ i, Measurable (Function.uncurry (f i))) :
     withDensity κ (∑' n, f n) = kernel.sum fun n => withDensity κ (f n) :=
@@ -146,7 +163,9 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
   congr with n
   rw [with_density_apply' _ (hf n) a hs]
 #align probability_theory.kernel.with_density_tsum ProbabilityTheory.kernel.withDensity_tsum
+-/
 
+#print ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded /-
 /-- If a kernel `κ` is finite and a function `f : α → β → ℝ≥0∞` is bounded, then `with_density κ f`
 is finite. -/
 theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKernel κ] {B : ℝ≥0∞}
@@ -166,7 +185,9 @@ theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKerne
   · rw [with_density_of_not_measurable _ hf]
     infer_instance
 #align probability_theory.kernel.is_finite_kernel_with_density_of_bounded ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded
+-/
 
+#print ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel /-
 /-- Auxiliary lemma for `is_s_finite_kernel_with_density`.
 If a kernel `κ` is finite, then `with_density κ f` is s-finite. -/
 theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFiniteKernel κ]
@@ -237,7 +258,9 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     _ ≤ n + 1 := (min_le_right _ _)
     _ = ↑(n + 1) := by norm_cast
 #align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
+-/
 
+#print ProbabilityTheory.kernel.IsSFiniteKernel.withDensity /-
 /-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
 `with_density κ f` is s-finite. -/
 theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel κ]
@@ -253,6 +276,7 @@ theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     is_s_finite_kernel_sum fun n =>
       is_s_finite_kernel_with_density_of_is_finite_kernel (seq κ n) hf_ne_top
 #align probability_theory.kernel.is_s_finite_kernel.with_density ProbabilityTheory.kernel.IsSFiniteKernel.withDensity
+-/
 
 /-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0`, `with_density κ f` is s-finite. -/
 instance (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0) :

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

@@ -42,8 +42,6 @@ namespace ProbabilityTheory.kernel
 
 variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
 
-include mα mβ
-
 variable {κ : kernel α β} {f : α → β → ℝ≥0∞}
 
 /-- Kernel with image `(κ a).with_density (f a)` if `function.uncurry f` is measurable, and

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -82,7 +82,7 @@ theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
 
 theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
-    (∫⁻ b, g b ∂withDensity κ f a) = ∫⁻ b, f a b * g b ∂κ a :=
+    ∫⁻ b, g b ∂withDensity κ f a = ∫⁻ b, f a b * g b ∂κ a :=
   by
   rw [kernel.with_density_apply _ hf,
     lintegral_with_density_eq_lintegral_mul _ (Measurable.of_uncurry_left hf) hg]
@@ -92,7 +92,7 @@ theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
 theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
     (hg : Measurable (Function.uncurry g)) :
-    (∫ b, f b ∂withDensity κ (fun a b => g a b) a) = ∫ b, g a b • f b ∂κ a :=
+    ∫ b, f b ∂withDensity κ (fun a b => g a b) a = ∫ b, g a b • f b ∂κ a :=
   by
   rw [kernel.with_density_apply, integral_withDensity_eq_integral_smul]
   · exact Measurable.of_uncurry_left hg
@@ -140,7 +140,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
       exact pi.summable.mpr fun p => ENNReal.summable
     rw [this]
     exact Measurable.ennreal_tsum' hf
-  have : (∫⁻ b in s, (∑' n, f n) a b ∂κ a) = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a :=
+  have : ∫⁻ b in s, (∑' n, f n) a b ∂κ a = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a :=
     by
     congr with b
     rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
@@ -161,7 +161,7 @@ theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKerne
           fun a => by
           rw [with_density_apply' κ hf a MeasurableSet.univ]
           calc
-            (∫⁻ b in Set.univ, f a b ∂κ a) ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
+            ∫⁻ b in Set.univ, f a b ∂κ a ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
             _ = B * κ a Set.univ := by
               simp only [measure.restrict_univ, MeasureTheory.lintegral_const]
             _ ≤ B * is_finite_kernel.bound κ := mul_le_mul_left' (measure_le_bound κ a Set.univ) _⟩⟩
@@ -211,7 +211,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     ext (a b) : 2
     rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
       ENNReal.tsum_eq_liminf_sum_nat]
-    have h_finset_sum : ∀ n, (∑ i in Finset.range n, fs i a b) = min (f a b) n :=
+    have h_finset_sum : ∀ n, ∑ i in Finset.range n, fs i a b = min (f a b) n :=
       by
       intro n
       induction' n with n hn

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -164,8 +164,7 @@ theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKerne
             (∫⁻ b in Set.univ, f a b ∂κ a) ≤ ∫⁻ b in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
             _ = B * κ a Set.univ := by
               simp only [measure.restrict_univ, MeasureTheory.lintegral_const]
-            _ ≤ B * is_finite_kernel.bound κ := mul_le_mul_left' (measure_le_bound κ a Set.univ) _
-            ⟩⟩
+            _ ≤ B * is_finite_kernel.bound κ := mul_le_mul_left' (measure_le_bound κ a Set.univ) _⟩⟩
   · rw [with_density_of_not_measurable _ hf]
     infer_instance
 #align probability_theory.kernel.is_finite_kernel_with_density_of_bounded ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded
@@ -239,7 +238,6 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     fs n a b ≤ min (f a b) (n + 1) := tsub_le_self
     _ ≤ n + 1 := (min_le_right _ _)
     _ = ↑(n + 1) := by norm_cast
-    
 #align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
 
 /-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,

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

@@ -70,8 +70,8 @@ protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) : withDensity κ f a = (κ a).withDensity (f a) :=
   by
   classical
-    rw [with_density, dif_pos hf]
-    rfl
+  rw [with_density, dif_pos hf]
+  rfl
 #align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
 
 theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]

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

@@ -186,7 +186,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     by
     intro a b n hn
     have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
-    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
+    rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this 
     · refine' this.trans (le_of_eq _)
       rw [ENNReal.ofReal_coe_nat]
     · norm_cast
@@ -207,7 +207,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
       suffices : ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0
       exact summable_of_ne_finset_zero this
       intro n hn_not_mem
-      rw [Finset.mem_range, not_lt] at hn_not_mem
+      rw [Finset.mem_range, not_lt] at hn_not_mem 
       exact h_zero a b n hn_not_mem
     ext (a b) : 2
     rw [tsum_apply (pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),

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

@@ -36,7 +36,7 @@ an s-finite kernel.
 
 open MeasureTheory ProbabilityTheory
 
-open MeasureTheory ENNReal NNReal BigOperators
+open scoped MeasureTheory ENNReal NNReal BigOperators
 
 namespace ProbabilityTheory.kernel
 

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

@@ -180,9 +180,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
   -- functions, and decompose an s-finite kernel as a sum of finite kernels. We then use that
   -- `with_density` commutes with sums for both arguments and get a sum of finite kernels.
   by_cases hf : Measurable (Function.uncurry f)
-  swap
-  · rw [with_density_of_not_measurable _ hf]
-    infer_instance
+  swap; · rw [with_density_of_not_measurable _ hf]; infer_instance
   let fs : ℕ → α → β → ℝ≥0∞ := fun n a b => min (f a b) (n + 1) - min (f a b) n
   have h_le : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ n :=
     by
@@ -232,13 +230,9 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     rw [Filter.EventuallyEq, Filter.eventually_atTop]
     exact ⟨⌈(f a b).toReal⌉₊, fun n hn => (min_eq_left (h_le a b n hn)).symm⟩
   rw [hf_eq_tsum, with_density_tsum _ fun n : ℕ => _]
-  swap
-  · exact (hf.min measurable_const).sub (hf.min measurable_const)
+  swap; · exact (hf.min measurable_const).sub (hf.min measurable_const)
   refine' is_s_finite_kernel_sum fun n => _
-  suffices is_finite_kernel (with_density κ (fs n))
-    by
-    haveI := this
-    infer_instance
+  suffices is_finite_kernel (with_density κ (fs n)) by haveI := this; infer_instance
   refine' is_finite_kernel_with_density_of_bounded _ (ENNReal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => _
   norm_cast
   calc

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -4,11 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.with_density
-! leanprover-community/mathlib commit 28b2a92f2996d28e580450863c130955de0ed398
+! leanprover-community/mathlib commit c0d694db494dd4f9aa57f2714b6e4c82b4ebc113
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Probability.Kernel.MeasurableIntegral
+import Mathbin.MeasureTheory.Integral.SetIntegral
 
 /-!
 # With Density
@@ -88,6 +89,16 @@ theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
   simp_rw [Pi.mul_apply]
 #align probability_theory.kernel.lintegral_with_density ProbabilityTheory.kernel.lintegral_withDensity
 
+theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    {f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
+    (hg : Measurable (Function.uncurry g)) :
+    (∫ b, f b ∂withDensity κ (fun a b => g a b) a) = ∫ b, g a b • f b ∂κ a :=
+  by
+  rw [kernel.with_density_apply, integral_withDensity_eq_integral_smul]
+  · exact Measurable.of_uncurry_left hg
+  · exact measurable_coe_nnreal_ennreal.comp hg
+#align probability_theory.kernel.integral_with_density ProbabilityTheory.kernel.integral_withDensity
+
 theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
     (f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f :=
   by

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

@@ -128,7 +128,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
       rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]
       exact pi.summable.mpr fun p => ENNReal.summable
     rw [this]
-    exact Measurable.eNNReal_tsum' hf
+    exact Measurable.ennreal_tsum' hf
   have : (∫⁻ b in s, (∑' n, f n) a b ∂κ a) = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a :=
     by
     congr with b

mathlib commit https://github.com/leanprover-community/mathlib/commit/28b2a92f2996d28e580450863c130955de0ed398

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 
 ! This file was ported from Lean 3 source module probability.kernel.with_density
-! leanprover-community/mathlib commit 483dd86cfec4a1380d22b1f6acd4c3dc53f501ff
+! leanprover-community/mathlib commit 28b2a92f2996d28e580450863c130955de0ed398
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -56,7 +56,7 @@ noncomputable def withDensity (κ : kernel α β) [IsSFiniteKernel κ] (f : α 
           property := by
             refine' measure.measurable_of_measurable_coe _ fun s hs => _
             simp_rw [with_density_apply _ hs]
-            exact measurable_set_lintegral κ hf hs } :
+            exact hf.set_lintegral_kernel_prod_right hs } :
         kernel α β))
     fun hf => 0
 #align probability_theory.kernel.with_density ProbabilityTheory.kernel.withDensity

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -260,7 +260,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
   norm_cast
   calc
     fs n a b ≤ min (f a b) (n + 1) := tsub_le_self
-    _ ≤ n + 1 := (min_le_right _ _)
+    _ ≤ n + 1 := min_le_right _ _
     _ = ↑(n + 1) := by norm_cast
 #align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
 

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -219,7 +219,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
     rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
     · refine' this.trans (le_of_eq _)
-      rw [ENNReal.ofReal_coe_nat]
+      rw [ENNReal.ofReal_natCast]
     · norm_cast
       exact zero_le _
   have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 := by

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)

@@ -38,7 +38,6 @@ open scoped MeasureTheory ENNReal NNReal BigOperators
 namespace ProbabilityTheory.kernel
 
 variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
-
 variable {κ : kernel α β} {f : α → β → ℝ≥0∞}
 
 /-- Kernel with image `(κ a).withDensity (f a)` if `Function.uncurry f` is measurable, and

@@ -150,7 +150,7 @@ lemma withDensity_add_right [IsSFiniteKernel κ] {f g : α → β → ℝ≥0∞
     withDensity κ (f + g) = withDensity κ f + withDensity κ g := by
   ext a
   rw [coeFn_add, Pi.add_apply, kernel.withDensity_apply _ hf, kernel.withDensity_apply _ hg,
-    kernel.withDensity_apply,Pi.add_apply, MeasureTheory.withDensity_add_right]
+    kernel.withDensity_apply, Pi.add_apply, MeasureTheory.withDensity_add_right]
   · exact hg.comp measurable_prod_mk_left
   · exact hf.add hg
 

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -226,7 +226,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
   have h_zero : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0 := by
     intro a b n hn
     suffices min (f a b) (n + 1) = f a b ∧ min (f a b) n = f a b by
-      simp_rw [this.1, this.2, tsub_self (f a b)]
+      simp_rw [fs, this.1, this.2, tsub_self (f a b)]
     exact ⟨min_eq_left ((h_le a b n hn).trans (le_add_of_nonneg_right zero_le_one)),
       min_eq_left (h_le a b n hn)⟩
   have hf_eq_tsum : f = ∑' n, fs n := by
@@ -245,7 +245,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
       induction' n with n hn
       · simp
       rw [Finset.sum_range_succ, hn]
-      simp
+      simp [fs]
     simp_rw [h_finset_sum]
     refine' (Filter.Tendsto.liminf_eq _).symm
     refine' Filter.Tendsto.congr' _ tendsto_const_nhds

Add a few simp lemmas about withDensity of constant functions 1 and 0, and the 3 more interesting lemmas

• withDensity_add_right: withDensity κ (f + g) = withDensity κ f + withDensity κ g
• withDensity_sub_add_cancel: withDensity κ (fun a x ↦ f a x - g a x) + withDensity κ g = withDensity κ f
• withDensity_mul: withDensity κ (fun a x ↦ f a x * g a x) = withDensity (withDensity κ fun a x ↦ f a x) g

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -72,6 +72,40 @@ protected theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
   rw [kernel.withDensity_apply κ hf, withDensity_apply' _ s]
 #align probability_theory.kernel.with_density_apply' ProbabilityTheory.kernel.withDensity_apply'
 
+nonrec lemma withDensity_congr_ae (κ : kernel α β) [IsSFiniteKernel κ] {f g : α → β → ℝ≥0∞}
+    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g))
+    (hfg : ∀ a, f a =ᵐ[κ a] g a) :
+    withDensity κ f = withDensity κ g := by
+  ext a
+  rw [kernel.withDensity_apply _ hf,kernel.withDensity_apply _ hg, withDensity_congr_ae (hfg a)]
+
+nonrec lemma withDensity_absolutelyContinuous [IsSFiniteKernel κ]
+    (f : α → β → ℝ≥0∞) (a : α) :
+    kernel.withDensity κ f a ≪ κ a := by
+  by_cases hf : Measurable (Function.uncurry f)
+  · rw [kernel.withDensity_apply _ hf]
+    exact withDensity_absolutelyContinuous _ _
+  · rw [withDensity_of_not_measurable _ hf]
+    simp [Measure.AbsolutelyContinuous.zero]
+
+@[simp]
+lemma withDensity_one (κ : kernel α β) [IsSFiniteKernel κ] :
+    kernel.withDensity κ 1 = κ := by
+  ext; rw [kernel.withDensity_apply _ measurable_const]; simp
+
+@[simp]
+lemma withDensity_one' (κ : kernel α β) [IsSFiniteKernel κ] :
+    kernel.withDensity κ (fun _ _ ↦ 1) = κ := kernel.withDensity_one _
+
+@[simp]
+lemma withDensity_zero (κ : kernel α β) [IsSFiniteKernel κ] :
+    kernel.withDensity κ 0 = 0 := by
+  ext; rw [kernel.withDensity_apply _ measurable_const]; simp
+
+@[simp]
+lemma withDensity_zero' (κ : kernel α β) [IsSFiniteKernel κ] :
+    kernel.withDensity κ (fun _ _ ↦ 0) = 0 := kernel.withDensity_zero _
+
 theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     (hf : Measurable (Function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
     ∫⁻ b, g b ∂withDensity κ f a = ∫⁻ b, f a b * g b ∂κ a := by
@@ -111,6 +145,25 @@ theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ :
     exact sum_zero.symm
 #align probability_theory.kernel.with_density_kernel_sum ProbabilityTheory.kernel.withDensity_kernel_sum
 
+lemma withDensity_add_right [IsSFiniteKernel κ] {f g : α → β → ℝ≥0∞}
+    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) :
+    withDensity κ (f + g) = withDensity κ f + withDensity κ g := by
+  ext a
+  rw [coeFn_add, Pi.add_apply, kernel.withDensity_apply _ hf, kernel.withDensity_apply _ hg,
+    kernel.withDensity_apply,Pi.add_apply, MeasureTheory.withDensity_add_right]
+  · exact hg.comp measurable_prod_mk_left
+  · exact hf.add hg
+
+lemma withDensity_sub_add_cancel [IsSFiniteKernel κ] {f g : α → β → ℝ≥0∞}
+    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g))
+    (hfg : ∀ a, g a ≤ᵐ[κ a] f a) :
+    withDensity κ (fun a x ↦ f a x - g a x) + withDensity κ g = withDensity κ f := by
+  rw [← withDensity_add_right _ hg]
+  swap; · exact hf.sub hg
+  refine withDensity_congr_ae κ ((hf.sub hg).add hg) hf (fun a ↦ ?_)
+  filter_upwards [hfg a] with x hx
+  rwa [Pi.add_apply, Pi.add_apply, tsub_add_cancel_iff_le]
+
 theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ] {f : ι → α → β → ℝ≥0∞}
     (hf : ∀ i, Measurable (Function.uncurry (f i))) :
     withDensity κ (∑' n, f n) = kernel.sum fun n => withDensity κ (f n) := by
@@ -230,4 +283,20 @@ instance (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0) :
     IsSFiniteKernel (withDensity κ fun a b => f a b) :=
   IsSFiniteKernel.withDensity κ fun _ _ => ENNReal.coe_ne_top
 
+nonrec lemma withDensity_mul [IsSFiniteKernel κ] {f : α → β → ℝ≥0} {g : α → β → ℝ≥0∞}
+    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) :
+    withDensity κ (fun a x ↦ f a x * g a x)
+      = withDensity (withDensity κ fun a x ↦ f a x) g := by
+  ext a : 1
+  rw [kernel.withDensity_apply]
+  swap; · exact (measurable_coe_nnreal_ennreal.comp hf).mul hg
+  change (Measure.withDensity (κ a) ((fun x ↦ (f a x : ℝ≥0∞)) * (fun x ↦ (g a x : ℝ≥0∞)))) =
+      (withDensity (withDensity κ fun a x ↦ f a x) g) a
+  rw [withDensity_mul]
+  · rw [kernel.withDensity_apply _ hg, kernel.withDensity_apply]
+    exact measurable_coe_nnreal_ennreal.comp hf
+  · rw [measurable_coe_nnreal_ennreal_iff]
+    exact hf.comp measurable_prod_mk_left
+  · exact hg.comp measurable_prod_mk_left
+
 end ProbabilityTheory.kernel

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -179,8 +179,8 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
   have hf_eq_tsum : f = ∑' n, fs n := by
     have h_sum_a : ∀ a, Summable fun n => fs n a := by
       refine' fun a => Pi.summable.mpr fun b => _
-      suffices : ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0
-      exact summable_of_ne_finset_zero this
+      suffices ∀ n, n ∉ Finset.range ⌈(f a b).toReal⌉₊ → fs n a b = 0 from
+        summable_of_ne_finset_zero this
       intro n hn_not_mem
       rw [Finset.mem_range, not_lt] at hn_not_mem
       exact h_zero a b n hn_not_mem

@sgouezel added a version of withDensity_apply that does not require measurability of the set if the measure is s-finite. This PR uses that result in other files of the library.

For results about rnDeriv, I put a prime on the version that assumes measurability of the set and no prime on the version for s-finite measures, as the second one should be the main use case.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: RemyDegenne <Remydegenne@gmail.com>

@@ -66,10 +66,10 @@ protected theorem withDensity_apply (κ : kernel α β) [IsSFiniteKernel κ]
   rfl
 #align probability_theory.kernel.with_density_apply ProbabilityTheory.kernel.withDensity_apply
 
-theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
-    (hf : Measurable (Function.uncurry f)) (a : α) {s : Set β} (hs : MeasurableSet s) :
+protected theorem withDensity_apply' (κ : kernel α β) [IsSFiniteKernel κ]
+    (hf : Measurable (Function.uncurry f)) (a : α) (s : Set β) :
     withDensity κ f a s = ∫⁻ b in s, f a b ∂κ a := by
-  rw [kernel.withDensity_apply κ hf, withDensity_apply _ hs]
+  rw [kernel.withDensity_apply κ hf, withDensity_apply' _ s]
 #align probability_theory.kernel.with_density_apply' ProbabilityTheory.kernel.withDensity_apply'
 
 theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
@@ -117,7 +117,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
   have h_sum_a : ∀ a, Summable fun n => f n a := fun a => Pi.summable.mpr fun b => ENNReal.summable
   have h_sum : Summable fun n => f n := Pi.summable.mpr h_sum_a
   ext a s hs
-  rw [sum_apply' _ a hs, withDensity_apply' κ _ a hs]
+  rw [sum_apply' _ a hs, kernel.withDensity_apply' κ _ a s]
   swap
   · have : Function.uncurry (∑' n, f n) = ∑' n, Function.uncurry (f n) := by
       ext1 p
@@ -131,7 +131,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
     rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
   rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).aemeasurable]
   congr with n
-  rw [withDensity_apply' _ (hf n) a hs]
+  rw [kernel.withDensity_apply' _ (hf n) a s]
 #align probability_theory.kernel.with_density_tsum ProbabilityTheory.kernel.withDensity_tsum
 
 /-- If a kernel `κ` is finite and a function `f : α → β → ℝ≥0∞` is bounded, then `withDensity κ f`
@@ -141,7 +141,7 @@ theorem isFiniteKernel_withDensity_of_bounded (κ : kernel α β) [IsFiniteKerne
   by_cases hf : Measurable (Function.uncurry f)
   · exact ⟨⟨B * IsFiniteKernel.bound κ, ENNReal.mul_lt_top hB_top (IsFiniteKernel.bound_ne_top κ),
       fun a => by
-        rw [withDensity_apply' κ hf a MeasurableSet.univ]
+        rw [kernel.withDensity_apply' κ hf a Set.univ]
         calc
           ∫⁻ b in Set.univ, f a b ∂κ a ≤ ∫⁻ _ in Set.univ, B ∂κ a := lintegral_mono (hf_B a)
           _ = B * κ a Set.univ := by

This incorporates changes from

• #7845
• #7847
• #7853
• #7872 (was never actually made to work, but the diffs in nightly-testing are unexciting: we need to fully qualify a few names)

They can all be closed when this is merged.

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

@@ -203,6 +203,8 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
   refine' isSFiniteKernel_sum fun n => _
   suffices IsFiniteKernel (withDensity κ (fs n)) by haveI := this; infer_instance
   refine' isFiniteKernel_withDensity_of_bounded _ (ENNReal.coe_ne_top : ↑n + 1 ≠ ∞) fun a b => _
+  -- After leanprover/lean4#2734, we need to do beta reduction before `norm_cast`
+  beta_reduce
   norm_cast
   calc
     fs n a b ≤ min (f a b) (n + 1) := tsub_le_self

Also add completeSpace_prod, integrable_prod.

@@ -80,7 +80,7 @@ theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
   simp_rw [Pi.mul_apply]
 #align probability_theory.kernel.lintegral_with_density ProbabilityTheory.kernel.lintegral_withDensity
 
-theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
     (hg : Measurable (Function.uncurry g)) :
     ∫ b, f b ∂withDensity κ (fun a b => g a b) a = ∫ b, g a b • f b ∂κ a := by

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

This has nice performance benefits.

@@ -37,7 +37,7 @@ open scoped MeasureTheory ENNReal NNReal BigOperators
 
 namespace ProbabilityTheory.kernel
 
-variable {α β ι : Type _} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
 
 variable {κ : kernel α β} {f : α → β → ℝ≥0∞}
 
@@ -80,7 +80,7 @@ theorem lintegral_withDensity (κ : kernel α β) [IsSFiniteKernel κ]
   simp_rw [Pi.mul_apply]
 #align probability_theory.kernel.lintegral_with_density ProbabilityTheory.kernel.lintegral_withDensity
 
-theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
     {f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
     (hg : Measurable (Function.uncurry g)) :
     ∫ b, f b ∂withDensity κ (fun a b => g a b) a = ∫ b, g a b • f b ∂κ a := by

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
-
-! This file was ported from Lean 3 source module probability.kernel.with_density
-! leanprover-community/mathlib commit c0d694db494dd4f9aa57f2714b6e4c82b4ebc113
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Probability.Kernel.MeasurableIntegral
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
+#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
+
 /-!
 # With Density
 

@@ -213,7 +213,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
     _ = ↑(n + 1) := by norm_cast
 #align probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel
 
-/-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
+/-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
 `withDensity κ f` is s-finite. -/
 nonrec theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel κ]
     (hf_ne_top : ∀ a b, f a b ≠ ∞) : IsSFiniteKernel (withDensity κ f) := by
@@ -226,7 +226,7 @@ nonrec theorem IsSFiniteKernel.withDensity (κ : kernel α β) [IsSFiniteKernel
     isSFiniteKernel_withDensity_of_isFiniteKernel (seq κ n) hf_ne_top
 #align probability_theory.kernel.is_s_finite_kernel.with_density ProbabilityTheory.kernel.IsSFiniteKernel.withDensity
 
-/-- For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0`, `withDensity κ f` is s-finite. -/
+/-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0`, `withDensity κ f` is s-finite. -/
 instance (κ : kernel α β) [IsSFiniteKernel κ] (f : α → β → ℝ≥0) :
     IsSFiniteKernel (withDensity κ fun a b => f a b) :=
   IsSFiniteKernel.withDensity κ fun _ _ => ENNReal.coe_ne_top

Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Pol'tta / Miyahara Kō <pol_tta@outlook.jp> Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Jireh Loreaux <loreaujy@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Heather Macbeth <25316162+hrmacbeth@users.noreply.github.com> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -187,7 +187,7 @@ theorem isSFiniteKernel_withDensity_of_isFiniteKernel (κ : kernel α β) [IsFin
       intro n hn_not_mem
       rw [Finset.mem_range, not_lt] at hn_not_mem
       exact h_zero a b n hn_not_mem
-    ext (a b) : 2
+    ext a b : 2
     rw [tsum_apply (Pi.summable.mpr h_sum_a), tsum_apply (h_sum_a a),
       ENNReal.tsum_eq_liminf_sum_nat]
     have h_finset_sum : ∀ n, ∑ i in Finset.range n, fs i a b = min (f a b) n := by

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -95,7 +95,7 @@ theorem integral_withDensity {E : Type _} [NormedAddCommGroup E] [NormedSpace 
 theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
     (f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f := by
   by_cases hf : Measurable (Function.uncurry f)
-  · ext (a s); intro
+  · ext a s
     simp only [kernel.withDensity_apply _ hf, coeFn_add, Pi.add_apply, withDensity_add_measure,
       Measure.add_apply]
   · simp_rw [withDensity_of_not_measurable _ hf]
@@ -119,7 +119,7 @@ theorem withDensity_tsum [Countable ι] (κ : kernel α β) [IsSFiniteKernel κ]
     withDensity κ (∑' n, f n) = kernel.sum fun n => withDensity κ (f n) := by
   have h_sum_a : ∀ a, Summable fun n => f n a := fun a => Pi.summable.mpr fun b => ENNReal.summable
   have h_sum : Summable fun n => f n := Pi.summable.mpr h_sum_a
-  ext (a s); intro hs
+  ext a s hs
   rw [sum_apply' _ a hs, withDensity_apply' κ _ a hs]
   swap
   · have : Function.uncurry (∑' n, f n) = ∑' n, Function.uncurry (f n) := by