mathlib documentation

measure_theory.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 also define integrable_on f s μ := integrable f (μ.restrict s) and prove theorems like integrable_on_union : integrable_on f (s ∪ t) μ ↔ integrable_on f s μ ∧ integrable_on f t μ.

Next we define a predicate integrable_at_filter (f : α → E) (l : filter α) (μ : measure α) saying that f is integrable at some set s ∈ l and prove that a measurable function is integrable at l with respect to μ provided that f is bounded above at l ⊓ μ.ae and μ is finite at 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/bochner_integration, but we reference them here because all theorems about set integrals are in this file.

TODO #

The file ends with over a hundred lines of commented out code. This is the old contents of this file using the indicator approach to the definition of ∫ x in s, f x ∂μ. This code should be migrated to the new definition.

theorem piecewise_ae_eq_restrict {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f g : α → β} (hs : measurable_set s) :
theorem piecewise_ae_eq_restrict_compl {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {s : set α} {f g : α → β} (hs : measurable_set s) :
theorem indicator_ae_eq_restrict {α : Type u_1} {β : Type u_2} [measurable_space α] [has_zero β] {μ : measure_theory.measure α} {s : set α} {f : α → β} (hs : measurable_set s) :
theorem indicator_ae_eq_restrict_compl {α : Type u_1} {β : Type u_2} [measurable_space α] [has_zero β] {μ : measure_theory.measure α} {s : set α} {f : α → β} (hs : measurable_set s) :
def measurable_at_filter {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (f : α → β) (l : filter α) (μ : measure_theory.measure α . "volume_tac") :
Prop

A function f is measurable at filter l w.r.t. a measure μ if it is ae-measurable w.r.t. μ.restrict s for some s ∈ l.

Equations
@[simp]
theorem measurable_at_bot {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {f : α → β} :
theorem measurable_at_filter.eventually {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {l : filter α} {f : α → β} {μ : measure_theory.measure α} (h : measurable_at_filter f l μ) :
∀ᶠ (s : set α) in l.lift' set.powerset, ae_measurable f (μ.restrict s)
theorem measurable_at_filter.filter_mono {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {l l' : filter α} {f : α → β} {μ : measure_theory.measure α} (h : measurable_at_filter f l μ) (h' : l' l) :
theorem ae_measurable.measurable_at_filter {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {l : filter α} {f : α → β} {μ : measure_theory.measure α} (h : ae_measurable f μ) :
theorem ae_measurable.measurable_at_filter_of_mem {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {l : filter α} {f : α → β} {μ : measure_theory.measure α} {s : set α} (h : ae_measurable f (μ.restrict s)) (hl : s l) :
theorem measurable.measurable_at_filter {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {l : filter α} {f : α → β} {μ : measure_theory.measure α} (h : measurable f) :
theorem measure_theory.has_finite_integral_restrict_of_bounded {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] {f : α → E} {s : set α} {μ : measure_theory.measure α} {C : } (hs : μ s < ) (hf : ∀ᵐ (x : α) ∂μ.restrict s, f x C) :
def measure_theory.integrable_on {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 . "volume_tac") :
Prop

A function is integrable_on a set s if it is a measurable function and if the integral of its pointwise norm over s is less than infinity.

Equations
theorem measure_theory.integrable_on.integrable {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} (h : measure_theory.integrable_on f s μ) :
@[simp]
theorem measure_theory.integrable_on_empty {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} :
theorem measure_theory.integrable_on_zero {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {s : set α} {μ : measure_theory.measure α} :
measure_theory.integrable_on (λ (_x : α), 0) s μ
theorem measure_theory.integrable_on_const {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {s : set α} {μ : measure_theory.measure α} {C : E} :
measure_theory.integrable_on (λ (_x : α), C) s μ C = 0 μ s <
theorem measure_theory.integrable_on.mono {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 ν : measure_theory.measure α} (h : measure_theory.integrable_on f t ν) (hs : s t) (hμ : μ ν) :
theorem measure_theory.integrable_on.mono_set {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} (h : measure_theory.integrable_on f t μ) (hst : s t) :
theorem measure_theory.integrable_on.mono_measure {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ ν : measure_theory.measure α} (h : measure_theory.integrable_on f s ν) (hμ : μ ν) :
theorem measure_theory.integrable_on.mono_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} (h : measure_theory.integrable_on f t μ) (hst : s ≤ᵐ[μ] t) :
theorem measure_theory.integrable.integrable_on {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} (h : measure_theory.integrable f μ) :
theorem measure_theory.integrable.integrable_on' {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} (h : measure_theory.integrable f (μ.restrict s)) :
theorem measure_theory.integrable_on.left_of_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} (h : measure_theory.integrable_on f (s t) μ) :
theorem measure_theory.integrable_on.right_of_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} (h : measure_theory.integrable_on f (s t) μ) :
theorem measure_theory.integrable_on.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} (hs : measure_theory.integrable_on f s μ) (ht : measure_theory.integrable_on f t μ) :
@[simp]
@[simp]
theorem measure_theory.integrable_on_finite_union {α : Type u_1} {β : Type u_2} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {μ : measure_theory.measure α} {s : set β} (hs : s.finite) {t : β → set α} :
measure_theory.integrable_on f (⋃ (i : β) (H : i s), t i) μ ∀ (i : β), i smeasure_theory.integrable_on f (t i) μ
@[simp]
theorem measure_theory.integrable_on_finset_union {α : Type u_1} {β : Type u_2} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {μ : measure_theory.measure α} {s : finset β} {t : β → set α} :
measure_theory.integrable_on f (⋃ (i : β) (H : i s), t i) μ ∀ (i : β), i smeasure_theory.integrable_on f (t i) μ
theorem measure_theory.integrable_on.add_measure {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] {f : α → E} {s : set α} {μ ν : measure_theory.measure α} (hμ : measure_theory.integrable_on f s μ) (hν : measure_theory.integrable_on f s ν) :
@[simp]
theorem measure_theory.ae_measurable_indicator_iff {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} (hs : measurable_set s) :
theorem measure_theory.integrable_on.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} (h : measure_theory.integrable_on f s μ) (hs : measurable_set s) :
theorem measure_theory.integrable.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} (h : measure_theory.integrable f μ) (hs : measurable_set s) :
def measure_theory.integrable_at_filter {α : Type u_1} {E : Type u_3} [measurable_space α] [normed_group E] [measurable_space E] (f : α → E) (l : filter α) (μ : measure_theory.measure α . "volume_tac") :
Prop

We say that a function f is integrable at filter l if it is integrable on some set s ∈ l. Equivalently, it is eventually integrable on s in l.lift' powerset.

Equations
theorem measure_theory.integrable_at_filter.eventually {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} {l : filter α} (h : measure_theory.integrable_at_filter f l μ) :
∀ᶠ (s : set α) in l.lift' set.powerset, measure_theory.integrable_on f s μ
theorem measure_theory.integrable_at_filter.filter_mono {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} {l l' : filter α} (hl : l l') (hl' : measure_theory.integrable_at_filter f l' μ) :

Alias of integrable_at_filter.inf_ae_iff.

If μ is a measure finite at filter l and f is a function such that its norm is bounded above at l, then f is integrable at l.

theorem filter.tendsto.integrable_at_filter_ae {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} {l : filter α} [l.is_measurably_generated] (hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l) {b : E} (hf : filter.tendsto f (l μ.ae) (𝓝 b)) :

Alias of measure.finite_at_filter.integrable_at_filter_of_tendsto_ae.

theorem filter.tendsto.integrable_at_filter {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} {l : filter α} [l.is_measurably_generated] (hfm : measurable_at_filter f l μ) (hμ : μ.finite_at_filter l) {b : E} (hf : filter.tendsto f l (𝓝 b)) :

Alias of measure.finite_at_filter.integrable_at_filter_of_tendsto.

theorem measure_theory.integrable.induction {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] (P : (α → E) → Prop) (h_ind : ∀ (c : E) ⦃s : set α⦄, measurable_set sμ s < P (s.indicator (λ (_x : α), c))) (h_add : ∀ ⦃f g : α → E⦄, disjoint (function.support f) (function.support g)measure_theory.integrable f μmeasure_theory.integrable g μP fP gP (f + g)) (h_closed : is_closed {f : (measure_theory.Lp E 1 μ) | P f}) (h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] gmeasure_theory.integrable f μP fP g) ⦃f : α → E⦄ (hf : measure_theory.integrable f μ) :
P f

To prove something for an arbitrary integrable function in a second countable Borel normed group, it suffices to show that

  • the property holds for (multiples of) characteristic functions;
  • is closed under addition;
  • the set of functions in the space for which the property holds is closed.
  • the property is closed under the almost-everywhere equal relation.

It is possible to make the hypotheses in the induction steps a bit stronger, and such conditions can be added once we need them (for example in h_add it is only necessary to consider the sum of a simple function with a multiple of a characteristic function and that the intersection of their images is a subset of {0}).

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.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_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).

@[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.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_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 {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_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 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.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.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 is_compact.integrable_on_of_nhds_within {α : Type u_1} {E : Type u_3} [measurable_space α] [measurable_space E] [normed_group E] [topological_space α] {μ : measure_theory.measure α} {s : set α} (hs : is_compact s) {f : α → E} (hf : ∀ (x : α), x smeasure_theory.integrable_at_filter f (𝓝[s] x) μ) :

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 continuous_on.ae_measurable {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} (hf : continuous_on f s) (hs : measurable_set s) :

A function which is continuous on a set s is almost everywhere measurable with respect to μ.restrict s.

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.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.

A function f continuous on a compact set s is integrable on this set with respect to any locally finite measure.

A continuous function f is integrable on any compact set with respect to any locally finite measure.

A continuous function with compact closure of the support is integrable on the whole space.

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.integrable_comp {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_space E] [normed_group F] [normed_space F] [measurable_space F] [borel_space F] [opens_measurable_space E] {φ : α → E} (L : E →L[] F) (φ_int : measure_theory.integrable φ μ) :
measure_theory.integrable (λ (a : α), L (φ a)) μ
theorem integral_of_real {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_2} [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_conj {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {𝕜 : Type u_2} [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_space E] [normed_group F] [normed_space F] [borel_space E] [topological_space.second_countable_topology E] [complete_space E] [measurable_space F] [borel_space F] [topological_space.second_countable_topology F] [complete_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} [normed_space E] [borel_space E] [topological_space.second_countable_topology E] [complete_space E] (f : α → ) (c : E) :
(x : α), f x c μ = ( (x : α), f x μ) c