mathlib documentation

measure_theory.integral.set_integral

Set integral #

In this file we prove some properties of ∫ x in s, f x ∂μ. Recall that this notation is defined as ∫ x, f x ∂(μ.restrict s). In integral_indicator we prove that for a measurable function f and a measurable set s this definition coincides with another natural definition: ∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ, where indicator s f x is equal to f x for x ∈ s and is zero otherwise.

Since ∫ x in s, f x ∂μ is a notation, one can rewrite or apply any theorem about ∫ x, f x ∂μ directly. In this file we prove some theorems about dependence of ∫ x in s, f x ∂μ on s, e.g. integral_union, integral_empty, integral_univ.

We use the property integrable_on f s μ := integrable f (μ.restrict s), defined in measure_theory.integrable_on. We also defined in that same file a predicate integrable_at_filter (f : α → E) (l : filter α) (μ : measure α) saying that f is integrable at some set s ∈ l.

Finally, we prove a version of the Fundamental theorem of calculus for set integral, see filter.tendsto.integral_sub_linear_is_o_ae and its corollaries. Namely, consider a measurably generated filter l, a measure μ finite at this filter, and a function f that has a finite limit c at l ⊓ μ.ae. Then ∫ x in s, f x ∂μ = μ s • c + o(μ s) as s tends to l.lift' powerset, i.e. for any ε>0 there exists t ∈ l such that ∥∫ x in s, f x ∂μ - μ s • c∥ ≤ ε * μ s whenever s ⊆ t. We also formulate a version of this theorem for a locally finite measure μ and a function f continuous at a point a.

Notation #

We provide the following notations for expressing the integral of a function on a set :

Note that the set notations are defined in the file measure_theory/integral/bochner, but we reference them here because all theorems about set integrals are in this file.

theorem measure_theory.set_integral_congr_ae {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f g : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hs : measurable_set s) (h : ∀ᵐ (x : α) ∂μ, x sf x = g x) :
(x : α) in s, f x μ = (x : α) in s, g x μ
theorem measure_theory.set_integral_congr {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f g : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hs : measurable_set s) (h : set.eq_on f g s) :
(x : α) in s, f x μ = (x : α) in s, g x μ
theorem measure_theory.set_integral_congr_set_ae {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hst : s =ᵐ[μ] t) :
(x : α) in s, f x μ = (x : α) in t, f x μ
theorem measure_theory.integral_union {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hst : disjoint s t) (hs : measurable_set s) (ht : measurable_set t) (hfs : measure_theory.integrable_on f s μ) (hft : measure_theory.integrable_on f t μ) :
(x : α) in s t, f x μ = (x : α) in s, f x μ + (x : α) in t, f x μ
theorem measure_theory.integral_union_ae {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hst : s t =ᵐ[μ] ) (hs : measurable_set s) (ht : measurable_set t) (hfs : measure_theory.integrable_on f s μ) (hft : measure_theory.integrable_on f t μ) :
(x : α) in s t, f x μ = (x : α) in s, f x μ + (x : α) in t, f x μ
theorem measure_theory.integral_diff {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hs : measurable_set s) (ht : measurable_set t) (hfs : measure_theory.integrable_on f s μ) (hft : measure_theory.integrable_on f t μ) (hts : t s) :
(x : α) in s \ t, f x μ = (x : α) in s, f x μ - (x : α) in t, f x μ
theorem measure_theory.integral_finset_bUnion {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {ι : Type u_2} (t : finset ι) {s : ι → set α} (hs : ∀ (i : ι), i tmeasurable_set (s i)) (h's : t.pairwise_on (disjoint on s)) (hf : ∀ (i : ι), i tmeasure_theory.integrable_on f (s i) μ) :
(x : α) in ⋃ (i : ι) (H : i t), s i, f x μ = ∑ (i : ι) in t, (x : α) in s i, f x μ
theorem measure_theory.integral_fintype_Union {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {ι : Type u_2} [fintype ι] {s : ι → set α} (hs : ∀ (i : ι), measurable_set (s i)) (h's : pairwise (disjoint on s)) (hf : ∀ (i : ι), measure_theory.integrable_on f (s i) μ) :
(x : α) in ⋃ (i : ι), s i, f x μ = ∑ (i : ι), (x : α) in s i, f x μ
theorem measure_theory.integral_univ {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] :
(x : α) in set.univ, f x μ = (x : α), f x μ
theorem measure_theory.integral_add_compl {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hs : measurable_set s) (hfi : measure_theory.integrable f μ) :
(x : α) in s, f x μ + (x : α) in s, f x μ = (x : α), f x μ
theorem measure_theory.integral_indicator {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (hs : measurable_set s) :
(x : α), s.indicator f x μ = (x : α) in s, f x μ

For a function f and a measurable set s, the integral of indicator s f over the whole space is equal to ∫ x in s, f x ∂μ defined as ∫ x, f x ∂(μ.restrict s).

theorem measure_theory.tendsto_set_integral_of_monotone {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {ι : Type u_2} [encodable ι] [semilattice_sup ι] {s : ι → set α} {f : α → E} (hsm : ∀ (i : ι), measurable_set (s i)) (h_mono : monotone s) (hfi : measure_theory.integrable_on f (⋃ (n : ι), s n) μ) :
filter.tendsto (λ (i : ι), (a : α) in s i, f a μ) filter.at_top (𝓝 ( (a : α) in ⋃ (n : ι), s n, f a μ))
theorem measure_theory.has_sum_integral_Union {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {ι : Type u_2} [encodable ι] {s : ι → set α} {f : α → E} (hm : ∀ (i : ι), measurable_set (s i)) (hd : pairwise (disjoint on s)) (hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
has_sum (λ (n : ι), (a : α) in s n, f a μ) ( (a : α) in ⋃ (n : ι), s n, f a μ)
theorem measure_theory.integral_Union {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {ι : Type u_2} [encodable ι] {s : ι → set α} {f : α → E} (hm : ∀ (i : ι), measurable_set (s i)) (hd : pairwise (disjoint on s)) (hfi : measure_theory.integrable_on f (⋃ (i : ι), s i) μ) :
(a : α) in ⋃ (n : ι), s n, f a μ = ∑' (n : ι), (a : α) in s n, f a μ
theorem measure_theory.set_integral_eq_zero_of_forall_eq_zero {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {f : α → E} (hf : measurable f) (ht_eq : ∀ (x : α), x tf x = 0) :
(x : α) in t, f x μ = 0
theorem measure_theory.set_integral_union_eq_left {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {s t : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {f : α → E} (hf : measurable f) (hfi : measure_theory.integrable f μ) (hs : measurable_set s) (ht : measurable_set t) (ht_eq : ∀ (x : α), x tf x = 0) :
(x : α) in s t, f x μ = (x : α) in s, f x μ
theorem measure_theory.set_integral_neg_eq_set_integral_nonpos {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] [linear_order E] [order_closed_topology E] {f : α → E} (hf : measurable f) (hfi : measure_theory.integrable f μ) :
(x : α) in {x : α | f x < 0}, f x μ = (x : α) in {x : α | f x 0}, f x μ
theorem measure_theory.integral_norm_eq_pos_sub_neg {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } (hf : measurable f) (hfi : measure_theory.integrable f μ) :
(x : α), f x μ = (x : α) in {x : α | 0 f x}, f x μ - (x : α) in {x : α | f x 0}, f x μ
@[simp]
theorem measure_theory.integral_indicator_const {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (e : E) ⦃s : set α⦄ (s_meas : measurable_set s) :
(a : α), s.indicator (λ (x : α), e) a μ = (μ s).to_real e
theorem measure_theory.set_integral_map {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {β : Type u_2} [measurable_space β] {g : α → β} {f : β → E} {s : set β} (hs : measurable_set s) (hf : ae_measurable f ((measure_theory.measure.map g) μ)) (hg : measurable g) :
(y : β) in s, f y (measure_theory.measure.map g) μ = (x : α) in g ⁻¹' s, f (g x) μ
theorem measure_theory.set_integral_map_of_closed_embedding {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] [topological_space α] [borel_space α] {β : Type u_2} [measurable_space β] [topological_space β] [borel_space β] {g : α → β} {f : β → E} {s : set β} (hs : measurable_set s) (hg : closed_embedding g) :
(y : β) in s, f y (measure_theory.measure.map g) μ = (x : α) in g ⁻¹' s, f (g x) μ
theorem measure_theory.set_integral_map_equiv {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {β : Type u_2} [measurable_space β] (e : α ≃ᵐ β) (f : β → E) (s : set β) :
(y : β) in s, f y (measure_theory.measure.map e) μ = (x : α) in e ⁻¹' s, f (e x) μ
theorem measure_theory.norm_set_integral_le_of_norm_le_const_ae {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {C : } (hs : μ s < ) (hC : ∀ᵐ (x : α) ∂μ.restrict s, f x C) :
(x : α) in s, f x μ C * (μ s).to_real
theorem measure_theory.norm_set_integral_le_of_norm_le_const_ae' {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {C : } (hs : μ s < ) (hC : ∀ᵐ (x : α) ∂μ, x sf x C) (hfm : ae_measurable f (μ.restrict s)) :
(x : α) in s, f x μ C * (μ s).to_real
theorem measure_theory.norm_set_integral_le_of_norm_le_const_ae'' {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {C : } (hs : μ s < ) (hsm : measurable_set s) (hC : ∀ᵐ (x : α) ∂μ, x sf x C) :
(x : α) in s, f x μ C * (μ s).to_real
theorem measure_theory.norm_set_integral_le_of_norm_le_const {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {C : } (hs : μ s < ) (hC : ∀ (x : α), x sf x C) (hfm : ae_measurable f (μ.restrict s)) :
(x : α) in s, f x μ C * (μ s).to_real
theorem measure_theory.norm_set_integral_le_of_norm_le_const' {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {C : } (hs : μ s < ) (hsm : measurable_set s) (hC : ∀ (x : α), x sf x C) :
(x : α) in s, f x μ C * (μ s).to_real
theorem measure_theory.set_integral_eq_zero_iff_of_nonneg_ae {α : Type u_1} [measurable_space α] {s : set α} {μ : measure_theory.measure α} {f : α → } (hf : 0 ≤ᵐ[μ.restrict s] f) (hfi : measure_theory.integrable_on f s μ) :
(x : α) in s, f x μ = 0 f =ᵐ[μ.restrict s] 0
theorem measure_theory.set_integral_pos_iff_support_of_nonneg_ae {α : Type u_1} [measurable_space α] {s : set α} {μ : measure_theory.measure α} {f : α → } (hf : 0 ≤ᵐ[μ.restrict s] f) (hfi : measure_theory.integrable_on f s μ) :
0 < (x : α) in s, f x μ 0 < μ (function.support f s)
theorem measure_theory.set_integral_trim {E : Type u_3} [normed_group E] [measurable_space E] [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {α : Type u_1} {m m0 : measurable_space α} {μ : measure_theory.measure α} (hm : m m0) {f : α → E} (hf_meas : measurable f) {s : set α} (hs : measurable_set s) :
(x : α) in s, f x μ = (x : α) in s, f x μ.trim hm
theorem measure_theory.set_integral_mono_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measure_theory.integrable_on f s μ) (hg : measure_theory.integrable_on g s μ) (h : f ≤ᵐ[μ.restrict s] g) :
(a : α) in s, f a μ (a : α) in s, g a μ
theorem measure_theory.set_integral_mono_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measure_theory.integrable_on f s μ) (hg : measure_theory.integrable_on g s μ) (h : f ≤ᵐ[μ] g) :
(a : α) in s, f a μ (a : α) in s, g a μ
theorem measure_theory.set_integral_mono_on {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measure_theory.integrable_on f s μ) (hg : measure_theory.integrable_on g s μ) (hs : measurable_set s) (h : ∀ (x : α), x sf x g x) :
(a : α) in s, f a μ (a : α) in s, g a μ
theorem measure_theory.set_integral_mono_on_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measure_theory.integrable_on f s μ) (hg : measure_theory.integrable_on g s μ) (hs : measurable_set s) (h : ∀ᵐ (x : α) ∂μ, x sf x g x) :
(a : α) in s, f a μ (a : α) in s, g a μ
theorem measure_theory.set_integral_mono {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measure_theory.integrable_on f s μ) (hg : measure_theory.integrable_on g s μ) (h : f g) :
(a : α) in s, f a μ (a : α) in s, g a μ
theorem measure_theory.set_integral_mono_set {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s t : set α} (hfi : measure_theory.integrable f μ) (hf : 0 ≤ᵐ[μ] f) (hst : s ≤ᵐ[μ] t) :
(x : α) in s, f x μ (x : α) in t, f x μ
theorem measure_theory.set_integral_nonneg_of_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hf : 0 ≤ᵐ[μ.restrict s] f) :
0 (a : α) in s, f a μ
theorem measure_theory.set_integral_nonneg_of_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hf : 0 ≤ᵐ[μ] f) :
0 (a : α) in s, f a μ
theorem measure_theory.set_integral_nonneg {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hs : measurable_set s) (hf : ∀ (a : α), a s0 f a) :
0 (a : α) in s, f a μ
theorem measure_theory.set_integral_nonneg_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hs : measurable_set s) (hf : ∀ᵐ (a : α) ∂μ, a s0 f a) :
0 (a : α) in s, f a μ
theorem measure_theory.set_integral_le_nonneg {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hs : measurable_set s) (hf : measurable f) (hfi : measure_theory.integrable f μ) :
(x : α) in s, f x μ (x : α) in {y : α | 0 f y}, f x μ
theorem measure_theory.set_integral_nonpos_of_ae_restrict {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hf : f ≤ᵐ[μ.restrict s] 0) :
(a : α) in s, f a μ 0
theorem measure_theory.set_integral_nonpos_of_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hf : f ≤ᵐ[μ] 0) :
(a : α) in s, f a μ 0
theorem measure_theory.set_integral_nonpos {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hs : measurable_set s) (hf : ∀ (a : α), a sf a 0) :
(a : α) in s, f a μ 0
theorem measure_theory.set_integral_nonpos_ae {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f : α → } {s : set α} (hs : measurable_set s) (hf : ∀ᵐ (a : α) ∂μ, a sf a 0) :
(a : α) in s, f a μ 0
theorem measure_theory.set_integral_nonpos_le {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {s : set α} (hs : measurable_set s) {f : α → } (hf : measurable f) (hfi : measure_theory.integrable f μ) :
(x : α) in {y : α | f y 0}, f x μ (x : α) in s, f x μ
theorem antitone.tendsto_set_integral {α : Type u_1} {E : Type u_3} [measurable_space α] {μ : measure_theory.measure α} [measurable_space E] [normed_group E] [borel_space E] [complete_space E] [normed_space E] [topological_space.second_countable_topology E] {s : set α} {f : α → E} (hsm : ∀ (i : ), measurable_set (s i)) (h_anti : antitone s) (hfi : measure_theory.integrable_on f (s 0) μ) :
filter.tendsto (λ (i : ), (a : α) in s i, f a μ) filter.at_top (𝓝 ( (a : α) in ⋂ (n : ), s n, f a μ))

Continuity of the set integral #

We prove that for any set s, the function λ f : α →₁[μ] E, ∫ x in s, f x ∂μ is continuous.

For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by (Lp.mem_ℒp f).restrict s).to_Lp f. This map is additive.

For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by (Lp.mem_ℒp f).restrict s).to_Lp f. This map commutes with scalar multiplication.

For f : Lp E p μ, we can define an element of Lp E p (μ.restrict s) by (Lp.mem_ℒp f).restrict s).to_Lp f. This map is non-expansive.

def measure_theory.Lp_to_Lp_restrict_clm (α : Type u_1) (F : Type u_4) [measurable_space α] (𝕜 : Type u_5) [is_R_or_C 𝕜] [measurable_space 𝕜] [normed_group F] [measurable_space F] [topological_space.second_countable_topology F] [borel_space F] [normed_space 𝕜 F] [borel_space 𝕜] (μ : measure_theory.measure α) (p : ℝ≥0∞) [hp : fact (1 p)] (s : set α) :

Continuous linear map sending a function of Lp F p μ to the same function in Lp F p (μ.restrict s).

Equations
theorem measure_theory.Lp_to_Lp_restrict_clm_coe_fn {α : Type u_1} {F : Type u_4} [measurable_space α] (𝕜 : Type u_5) [is_R_or_C 𝕜] [measurable_space 𝕜] [normed_group F] [measurable_space F] [topological_space.second_countable_topology F] [borel_space F] [normed_space 𝕜 F] {p : ℝ≥0∞} {μ : measure_theory.measure α} [borel_space 𝕜] [hp : fact (1 p)] (s : set α) (f : (measure_theory.Lp F p μ)) :
theorem filter.tendsto.integral_sub_linear_is_o_ae {α : Type u_1} {E : Type u_3} [measurable_space α] {ι : Type u_5} [measurable_space E] [normed_group E] [normed_space E] [topological_space.second_countable_topology E] [complete_space E] [borel_space E] {μ : measure_theory.measure α} {l : filter α} [l.is_measurably_generated] {f : α → E} {b : E} (h : filter.tendsto f (l μ.ae) (𝓝 b)) (hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l) {s : ι → set α} {li : filter ι} (hs : filter.tendsto s li (l.lift' set.powerset)) (m : ι → := λ (i : ι), (μ (s i)).to_real) (hsμ : (λ (i : ι), (μ (s i)).to_real) =ᶠ[li] m . "refl") :
asymptotics.is_o (λ (i : ι), (x : α) in s i, f x μ - m i b) m li

Fundamental theorem of calculus for set integrals: if μ is a measure that is finite at a filter l and f is a measurable function that has a finite limit b at l ⊓ μ.ae, then ∫ x in s i, f x ∂μ = μ (s i) • b + o(μ (s i)) at a filter li provided that s i tends to l.lift' powerset along li. Since μ (s i) is an ℝ≥0∞ number, we use (μ (s i)).to_real in the actual statement.

Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.

theorem continuous_within_at.integral_sub_linear_is_o_ae {α : Type u_1} {E : Type u_3} [measurable_space α] {ι : Type u_5} [measurable_space E] [normed_group E] [topological_space α] [opens_measurable_space α] [normed_space E] [topological_space.second_countable_topology E] [complete_space E] [borel_space E] {μ : measure_theory.measure α} [measure_theory.is_locally_finite_measure μ] {a : α} {t : set α} {f : α → E} (ha : continuous_within_at f t a) (ht : measurable_set t) (hfm : measurable_at_filter f (𝓝[t] a) μ) {s : ι → set α} {li : filter ι} (hs : filter.tendsto s li ((𝓝[t] a).lift' set.powerset)) (m : ι → := λ (i : ι), (μ (s i)).to_real) (hsμ : (λ (i : ι), (μ (s i)).to_real) =ᶠ[li] m . "refl") :
asymptotics.is_o (λ (i : ι), (x : α) in s i, f x μ - m i f a) m li

Fundamental theorem of calculus for set integrals, nhds_within version: if μ is a locally finite measure and f is an almost everywhere measurable function that is continuous at a point a within a measurable set t, then ∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i)) at a filter li provided that s i tends to (𝓝[t] a).lift' powerset along li. Since μ (s i) is an ℝ≥0∞ number, we use (μ (s i)).to_real in the actual statement.

Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.

theorem continuous_at.integral_sub_linear_is_o_ae {α : Type u_1} {E : Type u_3} [measurable_space α] {ι : Type u_5} [measurable_space E] [normed_group E] [topological_space α] [opens_measurable_space α] [normed_space E] [topological_space.second_countable_topology E] [complete_space E] [borel_space E] {μ : measure_theory.measure α} [measure_theory.is_locally_finite_measure μ] {a : α} {f : α → E} (ha : continuous_at f a) (hfm : measurable_at_filter f (𝓝 a) μ) {s : ι → set α} {li : filter ι} (hs : filter.tendsto s li ((𝓝 a).lift' set.powerset)) (m : ι → := λ (i : ι), (μ (s i)).to_real) (hsμ : (λ (i : ι), (μ (s i)).to_real) =ᶠ[li] m . "refl") :
asymptotics.is_o (λ (i : ι), (x : α) in s i, f x μ - m i f a) m li

Fundamental theorem of calculus for set integrals, nhds version: if μ is a locally finite measure and f is an almost everywhere measurable function that is continuous at a point a, then ∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i)) at li provided that s tends to (𝓝 a).lift' powerset along li. Sinceμ (s i)is anℝ≥0∞number, we use(μ (s i)).to_real` in the actual statement.

Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.

theorem continuous_on.measurable_at_filter {α : Type u_1} {β : Type u_2} [measurable_space α] [topological_space α] [opens_measurable_space α] [measurable_space β] [topological_space β] [borel_space β] {f : α → β} {s : set α} {μ : measure_theory.measure α} (hs : is_open s) (hf : continuous_on f s) (x : α) (H : x s) :

If a function is continuous on an open set s, then it is measurable at the filter 𝓝 x for all x ∈ s.

theorem continuous_at.measurable_at_filter {α : Type u_1} {E : Type u_3} [measurable_space α] [measurable_space E] [normed_group E] [topological_space α] [opens_measurable_space α] [borel_space E] {f : α → E} {s : set α} {μ : measure_theory.measure α} (hs : is_open s) (hf : ∀ (x : α), x scontinuous_at f x) (x : α) (H : x s) :
theorem continuous.measurable_at_filter {α : Type u_1} {β : Type u_2} [measurable_space α] [topological_space α] [opens_measurable_space α] [measurable_space β] [topological_space β] [borel_space β] {f : α → β} (hf : continuous f) (μ : measure_theory.measure α) (l : filter α) :
theorem continuous_on.measurable_at_filter_nhds_within {α : Type u_1} {β : Type u_2} [measurable_space α] [topological_space α] [opens_measurable_space α] [measurable_space β] [topological_space β] [borel_space β] {f : α → β} {s : set α} {μ : measure_theory.measure α} (hf : continuous_on f s) (hs : measurable_set s) (x : α) :

If a function is continuous on a measurable set s, then it is measurable at the filter 𝓝[s] x for all x.

theorem continuous_on.integral_sub_linear_is_o_ae {α : Type u_1} {E : Type u_3} [measurable_space α] {ι : Type u_5} [measurable_space E] [normed_group E] [topological_space α] [opens_measurable_space α] [normed_space E] [topological_space.second_countable_topology E] [complete_space E] [borel_space E] {μ : measure_theory.measure α} [measure_theory.is_locally_finite_measure μ] {a : α} {t : set α} {f : α → E} (hft : continuous_on f t) (ha : a t) (ht : measurable_set t) {s : ι → set α} {li : filter ι} (hs : filter.tendsto s li ((𝓝[t] a).lift' set.powerset)) (m : ι → := λ (i : ι), (μ (s i)).to_real) (hsμ : (λ (i : ι), (μ (s i)).to_real) =ᶠ[li] m . "refl") :
asymptotics.is_o (λ (i : ι), (x : α) in s i, f x μ - m i f a) m li

Fundamental theorem of calculus for set integrals, nhds_within version: if μ is a locally finite measure, f is continuous on a measurable set t, and a ∈ t, then ∫ x in (s i), f x ∂μ = μ (s i) • f a + o(μ (s i)) at li provided that s i tends to (𝓝[t] a).lift' powerset along li. Since μ (s i) is an ℝ≥0∞ number, we use (μ (s i)).to_real in the actual statement.

Often there is a good formula for (μ (s i)).to_real, so the formalization can take an optional argument m with this formula and a proof of(λ i, (μ (s i)).to_real) =ᶠ[li] m. Without these arguments,m i = (μ (s i)).to_real` is used in the output.

Continuous linear maps composed with integration #

The goal of this section is to prove that integration commutes with continuous linear maps. This holds for simple functions. The general result follows from the continuity of all involved operations on the space . Note that composition by a continuous linear map on is not just the composition, as we are dealing with classes of functions, but it has already been defined as continuous_linear_map.comp_Lp. We take advantage of this construction here.

theorem continuous_linear_map.integral_comp_Lp {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] {p : ℝ≥0∞} [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [borel_space E] [topological_space.second_countable_topology E] [normed_space F] (L : E →L[𝕜] F) (φ : (measure_theory.Lp E p μ)) :
(a : α), (L.comp_Lp φ) a μ = (a : α), L (φ a) μ
theorem continuous_linear_map.set_integral_comp_Lp {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] {p : ℝ≥0∞} [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [borel_space E] [topological_space.second_countable_topology E] [normed_space F] (L : E →L[𝕜] F) (φ : (measure_theory.Lp E p μ)) {s : set α} (hs : measurable_set s) :
(a : α) in s, (L.comp_Lp φ) a μ = (a : α) in s, L (φ a) μ
theorem continuous_linear_map.integral_comp_comm {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [borel_space E] [topological_space.second_countable_topology E] [normed_space F] [complete_space E] [measurable_space 𝕜] [opens_measurable_space 𝕜] [normed_space E] [is_scalar_tower 𝕜 E] [is_scalar_tower 𝕜 F] (L : E →L[𝕜] F) {φ : α → E} (φ_int : measure_theory.integrable φ μ) :
(a : α), L (φ a) μ = L ( (a : α), φ a μ)
theorem continuous_linear_map.integral_apply {α : Type u_1} {E : Type u_3} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] {H : Type u_2} [normed_group H] [normed_space H] [topological_space.second_countable_topology (H →L[] E)] {φ : α → (H →L[] E)} (φ_int : measure_theory.integrable φ μ) (v : H) :
( (a : α), φ a μ) v = (a : α), (φ a) v μ
theorem continuous_linear_map.integral_comp_comm' {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [borel_space E] [topological_space.second_countable_topology E] [normed_space F] [complete_space E] [measurable_space 𝕜] [opens_measurable_space 𝕜] [normed_space E] [is_scalar_tower 𝕜 E] [is_scalar_tower 𝕜 F] (L : E →L[𝕜] F) {K : ℝ≥0} (hL : antilipschitz_with K L) (φ : α → E) :
(a : α), L (φ a) μ = L ( (a : α), φ a μ)
theorem linear_isometry.integral_comp_comm {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [normed_space F] [is_scalar_tower 𝕜 F] [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] [is_scalar_tower 𝕜 E] [measurable_space 𝕜] [opens_measurable_space 𝕜] (L : E →ₗᵢ[𝕜] F) (φ : α → E) :
(a : α), L (φ a) μ = L ( (a : α), φ a μ)
theorem integral_of_real {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {f : α → } :
(a : α), (f a) μ = (a : α), f a μ
theorem integral_re {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {f : α → 𝕜} (hf : measure_theory.integrable f μ) :
(a : α), is_R_or_C.re (f a) μ = is_R_or_C.re ( (a : α), f a μ)
theorem integral_im {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {f : α → 𝕜} (hf : measure_theory.integrable f μ) :
(a : α), is_R_or_C.im (f a) μ = is_R_or_C.im ( (a : α), f a μ)
theorem integral_conj {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {f : α → 𝕜} :
(a : α), is_R_or_C.conj (f a) μ = is_R_or_C.conj ( (a : α), f a μ)
theorem integral_pair {α : Type u_1} {E : Type u_3} {F : Type u_4} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} [normed_group F] [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_space F] [normed_space F] {f : α → E} {g : α → F} (hf : measure_theory.integrable f μ) (hg : measure_theory.integrable g μ) :
(x : α), (f x, g x) μ = ( (x : α), f x μ, (x : α), g x μ)
theorem integral_smul_const {α : Type u_1} {E : Type u_3} [measurable_space α] [measurable_space E] [normed_group E] {μ : measure_theory.measure α} [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] (f : α → ) (c : E) :
(x : α), f x c μ = ( (x : α), f x μ) c
theorem integral_inner {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {E' : Type u_7} [inner_product_space 𝕜 E'] [measurable_space E'] [borel_space E'] [topological_space.second_countable_topology E'] [complete_space E'] [normed_space E'] [is_scalar_tower 𝕜 E'] {f : α → E'} (hf : measure_theory.integrable f μ) (c : E') :
(x : α), inner c (f x) μ = inner c ( (x : α), f x μ)
theorem integral_eq_zero_of_forall_integral_inner_eq_zero {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_6} [is_R_or_C 𝕜] [measurable_space 𝕜] [borel_space 𝕜] {E' : Type u_7} [inner_product_space 𝕜 E'] [measurable_space E'] [borel_space E'] [topological_space.second_countable_topology E'] [complete_space E'] [normed_space E'] [is_scalar_tower 𝕜 E'] (f : α → E') (hf : measure_theory.integrable f μ) (hf_int : ∀ (c : E'), (x : α), inner c (f x) μ = 0) :
(x : α), f x μ = 0