Measure spaces #
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The definition of a measure and a measure space are in measure_theory.measure_space_def, with
only a few basic properties. This file provides many more properties of these objects.
This separation allows the measurability tactic to import only the file measure_space_def, and to
be available in measure_space (through measurable_space).
Given a measurable space α, a measure on α is a function that sends measurable sets to the
extended nonnegative reals that satisfies the following conditions:
μ ∅ = 0;μis countably additive. This means that the measure of a countable union of pairwise disjoint sets is equal to the measure of the individual sets.
Every measure can be canonically extended to an outer measure, so that it assigns values to all subsets, not just the measurable subsets. On the other hand, a measure that is countably additive on measurable sets can be restricted to measurable sets to obtain a measure. In this file a measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.
Measures on α form a complete lattice, and are closed under scalar multiplication with ℝ≥0∞.
We introduce the following typeclasses for measures:
is_probability_measure μ:μ univ = 1;is_finite_measure μ:μ univ < ∞;sigma_finite μ: there exists a countable collection of sets that coverunivwhereμis finite;is_locally_finite_measure μ:∀ x, ∃ s ∈ 𝓝 x, μ s < ∞;has_no_atoms μ:∀ x, μ {x} = 0; possibly should be redefined as∀ s, 0 < μ s → ∃ t ⊆ s, 0 < μ t ∧ μ t < μ s.
Given a measure, the null sets are the sets where μ s = 0, where μ denotes the corresponding
outer measure (so s might not be measurable). We can then define the completion of μ as the
measure on the least σ-algebra that also contains all null sets, by defining the measure to be 0
on the null sets.
Main statements #
completionis the completion of a measure to all null measurable sets.measure.of_measurableandouter_measure.to_measureare two important ways to define a measure.
Implementation notes #
Given μ : measure α, μ s is the value of the outer measure applied to s.
This conveniently allows us to apply the measure to sets without proving that they are measurable.
We get countable subadditivity for all sets, but only countable additivity for measurable sets.
You often don't want to define a measure via its constructor. Two ways that are sometimes more convenient:
measure.of_measurableis a way to define a measure by only giving its value on measurable sets and proving the properties (1) and (2) mentioned above.outer_measure.to_measureis a way of obtaining a measure from an outer measure by showing that all measurable sets in the measurable space are Carathéodory measurable.
To prove that two measures are equal, there are multiple options:
ext: two measures are equal if they are equal on all measurable sets.ext_of_generate_from_of_Union: two measures are equal if they are equal on a π-system generating the measurable sets, if the π-system contains a spanning increasing sequence of sets where the measures take finite value (in particular the measures are σ-finite). This is a special case of the more generalext_of_generate_from_of_coverext_of_generate_finite: two finite measures are equal if they are equal on a π-system generating the measurable sets. This is a special case ofext_of_generate_from_of_UnionusingC ∪ {univ}, but is easier to work with.
A measure_space is a class that is a measurable space with a canonical measure.
The measure is denoted volume.
References #
- https://en.wikipedia.org/wiki/Measure_(mathematics)
- https://en.wikipedia.org/wiki/Complete_measure
- https://en.wikipedia.org/wiki/Almost_everywhere
Tags #
measure, almost everywhere, measure space, completion, null set, null measurable set
@[protected, instance]
def
measure_theory.ae_is_measurably_generated
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.ae_uIoc_iff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[linear_order α]
{a b : α}
{P : α → Prop} :
See also measure_theory.ae_restrict_uIoc_iff.
theorem
measure_theory.measure_union
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α}
(hd : disjoint s₁ s₂)
(h : measurable_set s₂) :
theorem
measure_theory.measure_union'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α}
(hd : disjoint s₁ s₂)
(h : measurable_set s₁) :
theorem
measure_theory.measure_inter_add_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measurable_set t) :
theorem
measure_theory.measure_diff_add_inter
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measurable_set t) :
theorem
measure_theory.measure_union_add_inter
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measurable_set t) :
theorem
measure_theory.measure_union_add_inter'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(t : set α) :
theorem
measure_theory.measure_add_measure_compl
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(h : measurable_set s) :
theorem
measure_theory.measure_bUnion₀
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set β}
{f : β → set α}
(hs : s.countable)
(hd : s.pairwise (measure_theory.ae_disjoint μ on f))
(h : ∀ (b : β), b ∈ s → measure_theory.null_measurable_set (f b) μ) :
theorem
measure_theory.measure_bUnion
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set β}
{f : β → set α}
(hs : s.countable)
(hd : s.pairwise_disjoint f)
(h : ∀ (b : β), b ∈ s → measurable_set (f b)) :
theorem
measure_theory.measure_sUnion₀
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{S : set (set α)}
(hs : S.countable)
(hd : S.pairwise (measure_theory.ae_disjoint μ))
(h : ∀ (s : set α), s ∈ S → measure_theory.null_measurable_set s μ) :
theorem
measure_theory.measure_bUnion_finset₀
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : finset ι}
{f : ι → set α}
(hd : ↑s.pairwise (measure_theory.ae_disjoint μ on f))
(hm : ∀ (b : ι), b ∈ s → measure_theory.null_measurable_set (f b) μ) :
theorem
measure_theory.measure_bUnion_finset
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : finset ι}
{f : ι → set α}
(hd : ↑s.pairwise_disjoint f)
(hm : ∀ (b : ι), b ∈ s → measurable_set (f b)) :
theorem
measure_theory.tsum_meas_le_meas_Union_of_disjoint
{α : Type u_1}
{ι : Type u_2}
[measurable_space α]
(μ : measure_theory.measure α)
{As : ι → set α}
(As_mble : ∀ (i : ι), measurable_set (As i))
(As_disj : pairwise (disjoint on As)) :
The measure of a disjoint union (even uncountable) of measurable sets is at least the sum of the measures of the sets.
theorem
measure_theory.tsum_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set β}
(hs : s.countable)
{f : α → β}
(hf : ∀ (y : β), y ∈ s → measurable_set (f ⁻¹' {y})) :
If s is a countable set, then the measure of its preimage can be found as the sum of measures
of the fibers f ⁻¹' {y}.
theorem
measure_theory.sum_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : finset β)
{f : α → β}
(hf : ∀ (y : β), y ∈ s → measurable_set (f ⁻¹' {y})) :
If s is a finset, then the measure of its preimage can be found as the sum of measures
of the fibers f ⁻¹' {y}.
theorem
measure_theory.measure_diff_null'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α}
(h : ⇑μ (s₁ ∩ s₂) = 0) :
theorem
measure_theory.measure_diff_null
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α}
(h : ⇑μ s₂ = 0) :
theorem
measure_theory.measure_add_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(t : set α) :
theorem
measure_theory.measure_diff'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(hm : measurable_set t)
(h_fin : ⇑μ t ≠ ⊤) :
theorem
measure_theory.measure_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α}
(h : s₂ ⊆ s₁)
(h₂ : measurable_set s₂)
(h_fin : ⇑μ s₂ ≠ ⊤) :
theorem
measure_theory.le_measure_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ : set α} :
theorem
measure_theory.measure_diff_lt_of_lt_add
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s)
(hst : s ⊆ t)
(hs' : ⇑μ s ≠ ⊤)
{ε : ennreal}
(h : ⇑μ t < ⇑μ s + ε) :
theorem
measure_theory.measure_diff_le_iff_le_add
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s)
(hst : s ⊆ t)
(hs' : ⇑μ s ≠ ⊤)
{ε : ennreal} :
theorem
measure_theory.measure_eq_measure_of_null_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hst : s ⊆ t)
(h_nulldiff : ⇑μ (t \ s) = 0) :
theorem
measure_theory.measure_eq_measure_smaller_of_between_null_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ s₃ : set α}
(h12 : s₁ ⊆ s₂)
(h23 : s₂ ⊆ s₃)
(h_nulldiff : ⇑μ (s₃ \ s₁) = 0) :
theorem
measure_theory.measure_eq_measure_larger_of_between_null_diff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s₁ s₂ s₃ : set α}
(h12 : s₁ ⊆ s₂)
(h23 : s₂ ⊆ s₃)
(h_nulldiff : ⇑μ (s₃ \ s₁) = 0) :
theorem
measure_theory.measure_compl
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(h₁ : measurable_set s)
(h_fin : ⇑μ s ≠ ⊤) :
@[simp]
theorem
measure_theory.union_ae_eq_left_iff_ae_subset
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α} :
@[simp]
theorem
measure_theory.union_ae_eq_right_iff_ae_subset
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α} :
theorem
measure_theory.ae_eq_of_ae_subset_of_measure_ge
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h₁ : s ≤ᵐ[μ] t)
(h₂ : ⇑μ t ≤ ⇑μ s)
(hsm : measurable_set s)
(ht : ⇑μ t ≠ ⊤) :
theorem
measure_theory.ae_eq_of_subset_of_measure_ge
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h₁ : s ⊆ t)
(h₂ : ⇑μ t ≤ ⇑μ s)
(hsm : measurable_set s)
(ht : ⇑μ t ≠ ⊤) :
If s ⊆ t, μ t ≤ μ s, μ t ≠ ∞, and s is measurable, then s =ᵐ[μ] t.
@[simp]
theorem
measure_theory.measure_Union_to_measurable
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable β]
(s : β → set α) :
theorem
measure_theory.measure_bUnion_to_measurable
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{I : set β}
(hc : I.countable)
(s : β → set α) :
@[simp]
theorem
measure_theory.measure_to_measurable_union
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α} :
@[simp]
theorem
measure_theory.measure_union_to_measurable
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α} :
theorem
measure_theory.sum_measure_le_measure_univ
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : finset ι}
{t : ι → set α}
(h : ∀ (i : ι), i ∈ s → measurable_set (t i))
(H : ↑s.pairwise_disjoint t) :
theorem
measure_theory.tsum_measure_le_measure_univ
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : ι → set α}
(hs : ∀ (i : ι), measurable_set (s i))
(H : pairwise (disjoint on s)) :
theorem
measure_theory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
(μ : measure_theory.measure α)
{s : ι → set α}
(hs : ∀ (i : ι), measurable_set (s i))
(H : ⇑μ set.univ < ∑' (i : ι), ⇑μ (s i)) :
Pigeonhole principle for measure spaces: if ∑' i, μ (s i) > μ univ, then
one of the intersections s i ∩ s j is not empty.
theorem
measure_theory.exists_nonempty_inter_of_measure_univ_lt_sum_measure
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
(μ : measure_theory.measure α)
{s : finset ι}
{t : ι → set α}
(h : ∀ (i : ι), i ∈ s → measurable_set (t i))
(H : ⇑μ set.univ < s.sum (λ (i : ι), ⇑μ (t i))) :
Pigeonhole principle for measure spaces: if s is a finset and
∑ i in s, μ (t i) > μ univ, then one of the intersections t i ∩ t j is not empty.
theorem
measure_theory.nonempty_inter_of_measure_lt_add
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{s t u : set α}
(ht : measurable_set t)
(h's : s ⊆ u)
(h't : t ⊆ u)
(h : ⇑μ u < ⇑μ s + ⇑μ t) :
If two sets s and t are included in a set u, and μ s + μ t > μ u,
then s intersects t. Version assuming that t is measurable.
theorem
measure_theory.nonempty_inter_of_measure_lt_add'
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{s t u : set α}
(hs : measurable_set s)
(h's : s ⊆ u)
(h't : t ⊆ u)
(h : ⇑μ u < ⇑μ s + ⇑μ t) :
If two sets s and t are included in a set u, and μ s + μ t > μ u,
then s intersects t. Version assuming that s is measurable.
theorem
measure_theory.measure_Union_eq_supr
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : directed has_subset.subset s) :
Continuity from below: the measure of the union of a directed sequence of (not necessarily -measurable) sets is the supremum of the measures.
theorem
measure_theory.measure_bUnion_eq_supr
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : ι → set α}
{t : set ι}
(ht : t.countable)
(hd : directed_on (has_subset.subset on s) t) :
theorem
measure_theory.measure_Inter_eq_infi
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(h : ∀ (i : ι), measurable_set (s i))
(hd : directed superset s)
(hfin : ∃ (i : ι), ⇑μ (s i) ≠ ⊤) :
Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the infimum of the measures.
theorem
measure_theory.tendsto_measure_Union
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
[semilattice_sup ι]
[countable ι]
{s : ι → set α}
(hm : monotone s) :
filter.tendsto (⇑μ ∘ s) filter.at_top (nhds (⇑μ (⋃ (n : ι), s n)))
Continuity from below: the measure of the union of an increasing sequence of measurable sets is the limit of the measures.
theorem
measure_theory.tendsto_measure_Inter
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
[semilattice_sup ι]
{s : ι → set α}
(hs : ∀ (n : ι), measurable_set (s n))
(hm : antitone s)
(hf : ∃ (i : ι), ⇑μ (s i) ≠ ⊤) :
filter.tendsto (⇑μ ∘ s) filter.at_top (nhds (⇑μ (⋂ (n : ι), s n)))
Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the limit of the measures.
theorem
measure_theory.tendsto_measure_bInter_gt
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{ι : Type u_2}
[linear_order ι]
[topological_space ι]
[order_topology ι]
[densely_ordered ι]
[topological_space.first_countable_topology ι]
{s : ι → set α}
{a : ι}
(hs : ∀ (r : ι), r > a → measurable_set (s r))
(hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j)
(hf : ∃ (r : ι) (H : r > a), ⇑μ (s r) ≠ ⊤) :
filter.tendsto (⇑μ ∘ s) (nhds_within a (set.Ioi a)) (nhds (⇑μ (⋂ (r : ι) (H : r > a), s r)))
The measure of the intersection of a decreasing sequence of measurable sets indexed by a linear order with first countable topology is the limit of the measures.
theorem
measure_theory.measure_limsup_eq_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : ℕ → set α}
(hs : ∑' (i : ℕ), ⇑μ (s i) ≠ ⊤) :
⇑μ (filter.limsup s filter.at_top) = 0
One direction of the Borel-Cantelli lemma: if (sᵢ) is a sequence of sets such
that ∑ μ sᵢ is finite, then the limit superior of the sᵢ is a null set.
theorem
measure_theory.measure_liminf_eq_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : ℕ → set α}
(h : ∑' (i : ℕ), ⇑μ (s i) ≠ ⊤) :
⇑μ (filter.liminf s filter.at_top) = 0
theorem
measure_theory.limsup_ae_eq_of_forall_ae_eq
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : ℕ → set α)
{t : set α}
(h : ∀ (n : ℕ), s n =ᵐ[μ] t) :
theorem
measure_theory.liminf_ae_eq_of_forall_ae_eq
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : ℕ → set α)
{t : set α}
(h : ∀ (n : ℕ), s n =ᵐ[μ] t) :
theorem
measure_theory.measure_if
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{x : β}
{t : set β}
{s : set α} :
noncomputable
def
measure_theory.outer_measure.to_measure
{α : Type u_1}
[ms : measurable_space α]
(m : measure_theory.outer_measure α)
(h : ms ≤ m.caratheodory) :
Obtain a measure by giving an outer measure where all sets in the σ-algebra are Carathéodory measurable.
Equations
- m.to_measure h = measure_theory.measure.of_measurable (λ (s : set α) (_x : measurable_set s), ⇑m s) _ _
theorem
measure_theory.le_to_outer_measure_caratheodory
{α : Type u_1}
[ms : measurable_space α]
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.to_measure_to_outer_measure
{α : Type u_1}
[ms : measurable_space α]
(m : measure_theory.outer_measure α)
(h : ms ≤ m.caratheodory) :
(m.to_measure h).to_outer_measure = m.trim
@[simp]
theorem
measure_theory.to_measure_apply
{α : Type u_1}
[ms : measurable_space α]
(m : measure_theory.outer_measure α)
(h : ms ≤ m.caratheodory)
{s : set α}
(hs : measurable_set s) :
⇑(m.to_measure h) s = ⇑m s
theorem
measure_theory.le_to_measure_apply
{α : Type u_1}
[ms : measurable_space α]
(m : measure_theory.outer_measure α)
(h : ms ≤ m.caratheodory)
(s : set α) :
⇑m s ≤ ⇑(m.to_measure h) s
theorem
measure_theory.to_measure_apply₀
{α : Type u_1}
[ms : measurable_space α]
(m : measure_theory.outer_measure α)
(h : ms ≤ m.caratheodory)
{s : set α}
(hs : measure_theory.null_measurable_set s (m.to_measure h)) :
⇑(m.to_measure h) s = ⇑m s
@[simp]
theorem
measure_theory.to_outer_measure_to_measure
{α : Type u_1}
[ms : measurable_space α]
{μ : measure_theory.measure α} :
μ.to_outer_measure.to_measure _ = μ
@[simp]
theorem
measure_theory.bounded_by_measure
{α : Type u_1}
[ms : measurable_space α]
(μ : measure_theory.measure α) :
theorem
measure_theory.measure.measure_inter_eq_of_measure_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t u : set α}
(hs : measurable_set s)
(h : ⇑μ t = ⇑μ u)
(htu : t ⊆ u)
(ht_ne_top : ⇑μ t ≠ ⊤) :
If u is a superset of t with the same (finite) measure (both sets possibly non-measurable),
then for any measurable set s one also has μ (t ∩ s) = μ (u ∩ s).
theorem
measure_theory.measure.measure_to_measurable_inter
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s)
(ht : ⇑μ t ≠ ⊤) :
The measurable superset to_measurable μ t of t (which has the same measure as t)
satisfies, for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (u ∩ s).
Here, we require that the measure of t is finite. The conclusion holds without this assumption
when the measure is sigma_finite, see measure_to_measurable_inter_of_sigma_finite.
The ℝ≥0∞-module of measures #
@[protected, instance]
Equations
- measure_theory.measure.has_zero = {zero := {to_outer_measure := 0, m_Union := _, trimmed := _}}
@[simp]
theorem
measure_theory.measure.zero_to_outer_measure
{α : Type u_1}
{m : measurable_space α} :
0.to_outer_measure = 0
@[simp, norm_cast]
@[protected, instance]
theorem
measure_theory.measure.eq_zero_of_is_empty
{α : Type u_1}
[is_empty α]
{m : measurable_space α}
(μ : measure_theory.measure α) :
μ = 0
@[protected, instance]
Equations
@[protected, instance]
Equations
- measure_theory.measure.has_add = {add := λ (μ₁ μ₂ : measure_theory.measure α), {to_outer_measure := μ₁.to_outer_measure + μ₂.to_outer_measure, m_Union := _, trimmed := _}}
@[simp]
theorem
measure_theory.measure.add_to_outer_measure
{α : Type u_1}
{m : measurable_space α}
(μ₁ μ₂ : measure_theory.measure α) :
(μ₁ + μ₂).to_outer_measure = μ₁.to_outer_measure + μ₂.to_outer_measure
@[simp, norm_cast]
theorem
measure_theory.measure.coe_add
{α : Type u_1}
{m : measurable_space α}
(μ₁ μ₂ : measure_theory.measure α) :
theorem
measure_theory.measure.add_apply
{α : Type u_1}
{m : measurable_space α}
(μ₁ μ₂ : measure_theory.measure α)
(s : set α) :
@[protected, instance]
def
measure_theory.measure.has_smul
{α : Type u_1}
{R : Type u_6}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
[measurable_space α] :
Equations
- measure_theory.measure.has_smul = {smul := λ (c : R) (μ : measure_theory.measure α), {to_outer_measure := c • μ.to_outer_measure, m_Union := _, trimmed := _}}
@[simp]
theorem
measure_theory.measure.smul_to_outer_measure
{α : Type u_1}
{R : Type u_6}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
{m : measurable_space α}
(c : R)
(μ : measure_theory.measure α) :
(c • μ).to_outer_measure = c • μ.to_outer_measure
@[simp, norm_cast]
theorem
measure_theory.measure.coe_smul
{α : Type u_1}
{R : Type u_6}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
{m : measurable_space α}
(c : R)
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.measure.smul_apply
{α : Type u_1}
{R : Type u_6}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
{m : measurable_space α}
(c : R)
(μ : measure_theory.measure α)
(s : set α) :
@[protected, instance]
def
measure_theory.measure.smul_comm_class
{α : Type u_1}
{R : Type u_6}
{R' : Type u_7}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
[has_smul R' ennreal]
[is_scalar_tower R' ennreal ennreal]
[smul_comm_class R R' ennreal]
[measurable_space α] :
smul_comm_class R R' (measure_theory.measure α)
@[protected, instance]
def
measure_theory.measure.is_scalar_tower
{α : Type u_1}
{R : Type u_6}
{R' : Type u_7}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
[has_smul R' ennreal]
[is_scalar_tower R' ennreal ennreal]
[has_smul R R']
[is_scalar_tower R R' ennreal]
[measurable_space α] :
is_scalar_tower R R' (measure_theory.measure α)
@[protected, instance]
def
measure_theory.measure.is_central_scalar
{α : Type u_1}
{R : Type u_6}
[has_smul R ennreal]
[is_scalar_tower R ennreal ennreal]
[has_smul Rᵐᵒᵖ ennreal]
[is_central_scalar R ennreal]
[measurable_space α] :
@[protected, instance]
def
measure_theory.measure.mul_action
{α : Type u_1}
{R : Type u_6}
[monoid R]
[mul_action R ennreal]
[is_scalar_tower R ennreal ennreal]
[measurable_space α] :
Equations
- measure_theory.measure.mul_action = function.injective.mul_action measure_theory.measure.to_outer_measure measure_theory.measure.to_outer_measure_injective measure_theory.measure.mul_action._proof_1
@[protected, instance]
Equations
- measure_theory.measure.add_comm_monoid = function.injective.add_comm_monoid measure_theory.measure.to_outer_measure measure_theory.measure.to_outer_measure_injective measure_theory.measure.zero_to_outer_measure measure_theory.measure.add_to_outer_measure measure_theory.measure.add_comm_monoid._proof_2
Coercion to function as an additive monoid homomorphism.
Equations
@[simp]
theorem
measure_theory.measure.coe_finset_sum
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
(I : finset ι)
(μ : ι → measure_theory.measure α) :
theorem
measure_theory.measure.finset_sum_apply
{α : Type u_1}
{ι : Type u_5}
{m : measurable_space α}
(I : finset ι)
(μ : ι → measure_theory.measure α)
(s : set α) :
@[protected, instance]
noncomputable
def
measure_theory.measure.distrib_mul_action
{α : Type u_1}
{R : Type u_6}
[monoid R]
[distrib_mul_action R ennreal]
[is_scalar_tower R ennreal ennreal]
[measurable_space α] :
Equations
- measure_theory.measure.distrib_mul_action = function.injective.distrib_mul_action {to_fun := measure_theory.measure.to_outer_measure _inst_6, map_zero' := _, map_add' := _} measure_theory.measure.to_outer_measure_injective measure_theory.measure.distrib_mul_action._proof_1
@[protected, instance]
noncomputable
def
measure_theory.measure.module
{α : Type u_1}
{R : Type u_6}
[semiring R]
[module R ennreal]
[is_scalar_tower R ennreal ennreal]
[measurable_space α] :
module R (measure_theory.measure α)
Equations
- measure_theory.measure.module = function.injective.module R {to_fun := measure_theory.measure.to_outer_measure _inst_6, map_zero' := _, map_add' := _} measure_theory.measure.to_outer_measure_injective measure_theory.measure.module._proof_1
@[simp]
theorem
measure_theory.measure.coe_nnreal_smul_apply
{α : Type u_1}
{m : measurable_space α}
(c : nnreal)
(μ : measure_theory.measure α)
(s : set α) :
theorem
measure_theory.measure.measure_to_measurable_add_inter_left
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s)
(ht : ⇑(μ + ν) t ≠ ⊤) :
theorem
measure_theory.measure.measure_to_measurable_add_inter_right
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s)
(ht : ⇑(μ + ν) t ≠ ⊤) :
The complete lattice of measures #
@[protected, instance]
Measures are partially ordered.
The definition of less equal here is equivalent to the definition without the
measurable set condition, and this is shown by measure.le_iff'. It is defined
this way since, to prove μ ≤ ν, we may simply intros s hs instead of rewriting followed
by intros s hs.
Equations
- measure_theory.measure.partial_order = {le := λ (m₁ m₂ : measure_theory.measure α), ∀ (s : set α), measurable_set s → ⇑m₁ s ≤ ⇑m₂ s, lt := preorder.lt._default (λ (m₁ m₂ : measure_theory.measure α), ∀ (s : set α), measurable_set s → ⇑m₁ s ≤ ⇑m₂ s), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _}
theorem
measure_theory.measure.le_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ₁ μ₂ : measure_theory.measure α} :
theorem
measure_theory.measure.to_outer_measure_le
{α : Type u_1}
{m0 : measurable_space α}
{μ₁ μ₂ : measure_theory.measure α} :
μ₁.to_outer_measure ≤ μ₂.to_outer_measure ↔ μ₁ ≤ μ₂
theorem
measure_theory.measure.le_iff'
{α : Type u_1}
{m0 : measurable_space α}
{μ₁ μ₂ : measure_theory.measure α} :
theorem
measure_theory.measure.lt_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
theorem
measure_theory.measure.lt_iff'
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
@[protected, instance]
@[protected]
theorem
measure_theory.measure.le_add_left
{α : Type u_1}
{m0 : measurable_space α}
{μ ν ν' : measure_theory.measure α}
(h : μ ≤ ν) :
@[protected]
theorem
measure_theory.measure.le_add_right
{α : Type u_1}
{m0 : measurable_space α}
{μ ν ν' : measure_theory.measure α}
(h : μ ≤ ν) :
theorem
measure_theory.measure.Inf_caratheodory
{α : Type u_1}
{m0 : measurable_space α}
{m : set (measure_theory.measure α)}
(s : set α)
(hs : measurable_set s) :
@[protected, instance]
Equations
theorem
measure_theory.measure.Inf_apply
{α : Type u_1}
{m0 : measurable_space α}
{s : set α}
{m : set (measure_theory.measure α)}
(hs : measurable_set s) :
⇑(has_Inf.Inf m) s = ⇑(has_Inf.Inf (measure_theory.measure.to_outer_measure '' m)) s
@[protected, instance]
noncomputable
def
measure_theory.measure.complete_semilattice_Inf
{α : Type u_1}
[measurable_space α] :
Equations
- measure_theory.measure.complete_semilattice_Inf = {le := partial_order.le measure_theory.measure.partial_order, lt := partial_order.lt measure_theory.measure.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, Inf := has_Inf.Inf measure_theory.measure.has_Inf, Inf_le := _, le_Inf := _}
@[protected, instance]
Equations
- measure_theory.measure.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), le := complete_lattice.le (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), lt := complete_lattice.lt (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := complete_lattice.inf (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), Inf_le := _, le_Inf := _, top := complete_lattice.top (complete_lattice_of_complete_semilattice_Inf (measure_theory.measure α)), bot := 0, le_top := _, bot_le := _}
@[simp]
theorem
measure_theory.outer_measure.to_measure_top
{α : Type u_1}
[measurable_space α] :
⊤.to_measure _ = ⊤
@[simp]
@[simp]
theorem
measure_theory.measure.top_add
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.measure.add_top
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[protected]
theorem
measure_theory.measure.zero_le
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
0 ≤ μ
theorem
measure_theory.measure.nonpos_iff_eq_zero'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.measure.measure_univ_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.measure.measure_univ_ne_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.measure.measure_univ_pos
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
Pushforward and pullback #
noncomputable
def
measure_theory.measure.lift_linear
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{m0 : measurable_space α}
(f : measure_theory.outer_measure α →ₗ[ennreal] measure_theory.outer_measure β)
(hf : ∀ (μ : measure_theory.measure α), _inst_1 ≤ (⇑f μ.to_outer_measure).caratheodory) :
Lift a linear map between outer_measure spaces such that for each measure μ every measurable
set is caratheodory-measurable w.r.t. f μ to a linear map between measure spaces.
Equations
- measure_theory.measure.lift_linear f hf = {to_fun := λ (μ : measure_theory.measure α), (⇑f μ.to_outer_measure).to_measure _, map_add' := _, map_smul' := _}
@[simp]
theorem
measure_theory.measure.lift_linear_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : measure_theory.outer_measure α →ₗ[ennreal] measure_theory.outer_measure β}
(hf : ∀ (μ : measure_theory.measure α), _inst_1 ≤ (⇑f μ.to_outer_measure).caratheodory)
{s : set β}
(hs : measurable_set s) :
⇑(⇑(measure_theory.measure.lift_linear f hf) μ) s = ⇑(⇑f μ.to_outer_measure) s
theorem
measure_theory.measure.le_lift_linear_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : measure_theory.outer_measure α →ₗ[ennreal] measure_theory.outer_measure β}
(hf : ∀ (μ : measure_theory.measure α), _inst_1 ≤ (⇑f μ.to_outer_measure).caratheodory)
(s : set β) :
⇑(⇑f μ.to_outer_measure) s ≤ ⇑(⇑(measure_theory.measure.lift_linear f hf) μ) s
noncomputable
def
measure_theory.measure.mapₗ
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
(f : α → β) :
The pushforward of a measure as a linear map. It is defined to be 0 if f is not
a measurable function.
Equations
- measure_theory.measure.mapₗ f = dite (measurable f) (λ (hf : measurable f), measure_theory.measure.lift_linear (measure_theory.outer_measure.map f) _) (λ (hf : ¬measurable f), 0)
theorem
measure_theory.measure.mapₗ_congr
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f g : α → β}
(hf : measurable f)
(hg : measurable g)
(h : f =ᵐ[μ] g) :
@[irreducible]
noncomputable
def
measure_theory.measure.map
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
(f : α → β)
(μ : measure_theory.measure α) :
The pushforward of a measure. It is defined to be 0 if f is not an almost everywhere
measurable function.
Equations
- measure_theory.measure.map f μ = dite (ae_measurable f μ) (λ (hf : ae_measurable f μ), ⇑(measure_theory.measure.mapₗ (ae_measurable.mk f hf)) μ) (λ (hf : ¬ae_measurable f μ), 0)
theorem
measure_theory.measure.mapₗ_mk_apply_of_ae_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ) :
⇑(measure_theory.measure.mapₗ (ae_measurable.mk f hf)) μ = measure_theory.measure.map f μ
theorem
measure_theory.measure.mapₗ_apply_of_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{f : α → β}
(hf : measurable f)
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.measure.map_add
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
(μ ν : measure_theory.measure α)
{f : α → β}
(hf : measurable f) :
measure_theory.measure.map f (μ + ν) = measure_theory.measure.map f μ + measure_theory.measure.map f ν
@[simp]
theorem
measure_theory.measure.map_zero
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
(f : α → β) :
measure_theory.measure.map f 0 = 0
theorem
measure_theory.measure.map_of_not_ae_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{f : α → β}
{μ : measure_theory.measure α}
(hf : ¬ae_measurable f μ) :
measure_theory.measure.map f μ = 0
theorem
measure_theory.measure.map_congr
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f g : α → β}
(h : f =ᵐ[μ] g) :
@[protected, simp]
theorem
measure_theory.measure.map_smul
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
(c : ennreal)
(μ : measure_theory.measure α)
(f : α → β) :
measure_theory.measure.map f (c • μ) = c • measure_theory.measure.map f μ
@[protected, simp]
theorem
measure_theory.measure.map_smul_nnreal
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
(c : nnreal)
(μ : measure_theory.measure α)
(f : α → β) :
measure_theory.measure.map f (c • μ) = c • measure_theory.measure.map f μ
@[simp]
theorem
measure_theory.measure.map_apply_of_ae_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{s : set β}
(hs : measurable_set s) :
⇑(measure_theory.measure.map f μ) s = ⇑μ (f ⁻¹' s)
We can evaluate the pushforward on measurable sets. For non-measurable sets, see
measure_theory.measure.le_map_apply and measurable_equiv.map_apply.
@[simp]
theorem
measure_theory.measure.map_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : measurable f)
{s : set β}
(hs : measurable_set s) :
⇑(measure_theory.measure.map f μ) s = ⇑μ (f ⁻¹' s)
theorem
measure_theory.measure.map_to_outer_measure
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ) :
@[simp]
theorem
measure_theory.measure.map_id
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.measure.map_id'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
measure_theory.measure.map (λ (x : α), x) μ = μ
theorem
measure_theory.measure.map_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m0 : measurable_space α}
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{g : β → γ}
{f : α → β}
(hg : measurable g)
(hf : measurable f) :
measure_theory.measure.map g (measure_theory.measure.map f μ) = measure_theory.measure.map (g ∘ f) μ
theorem
measure_theory.measure.map_mono
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ ν : measure_theory.measure α}
{f : α → β}
(h : μ ≤ ν)
(hf : measurable f) :
theorem
measure_theory.measure.le_map_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
(s : set β) :
⇑μ (f ⁻¹' s) ≤ ⇑(measure_theory.measure.map f μ) s
Even if s is not measurable, we can bound map f μ s from below.
See also measurable_equiv.map_apply.
theorem
measure_theory.measure.preimage_null_of_map_null
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{s : set β}
(hs : ⇑(measure_theory.measure.map f μ) s = 0) :
Even if s is not measurable, map f μ s = 0 implies that μ (f ⁻¹' s) = 0.
theorem
measure_theory.measure.tendsto_ae_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ) :
filter.tendsto f μ.ae (measure_theory.measure.map f μ).ae
noncomputable
def
measure_theory.measure.comapₗ
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
(f : α → β) :
Pullback of a measure as a linear map. If f sends each measurable set to a measurable
set, then for each measurable set s we have comapₗ f μ s = μ (f '' s).
If the linearity is not needed, please use comap instead, which works for a larger class of
functions.
Equations
- measure_theory.measure.comapₗ f = dite (function.injective f ∧ ∀ (s : set α), measurable_set s → measurable_set (f '' s)) (λ (hf : function.injective f ∧ ∀ (s : set α), measurable_set s → measurable_set (f '' s)), measure_theory.measure.lift_linear (measure_theory.outer_measure.comap f) _) (λ (hf : ¬(function.injective f ∧ ∀ (s : set α), measurable_set s → measurable_set (f '' s))), 0)
theorem
measure_theory.measure.comapₗ_apply
{α : Type u_1}
{s : set α}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measurable_set (f '' s))
(μ : measure_theory.measure β)
(hs : measurable_set s) :
noncomputable
def
measure_theory.measure.comap
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
(f : α → β)
(μ : measure_theory.measure β) :
Pullback of a measure. If f sends each measurable set to a null-measurable set,
then for each measurable set s we have comap f μ s = μ (f '' s).
Equations
- measure_theory.measure.comap f μ = dite (function.injective f ∧ ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ) (λ (hf : function.injective f ∧ ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ), (⇑(measure_theory.outer_measure.comap f) μ.to_outer_measure).to_measure _) (λ (hf : ¬(function.injective f ∧ ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)), 0)
theorem
measure_theory.measure.comap_apply₀
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{s : set α}
[measurable_space α]
(f : α → β)
(μ : measure_theory.measure β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)
(hs : measure_theory.null_measurable_set s (measure_theory.measure.comap f μ)) :
⇑(measure_theory.measure.comap f μ) s = ⇑μ (f '' s)
theorem
measure_theory.measure.le_comap_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(μ : measure_theory.measure β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)
(s : set α) :
⇑μ (f '' s) ≤ ⇑(measure_theory.measure.comap f μ) s
theorem
measure_theory.measure.comap_apply
{α : Type u_1}
{s : set α}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measurable_set (f '' s))
(μ : measure_theory.measure β)
(hs : measurable_set s) :
⇑(measure_theory.measure.comap f μ) s = ⇑μ (f '' s)
theorem
measure_theory.measure.comapₗ_eq_comap
{α : Type u_1}
{s : set α}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measurable_set (f '' s))
(μ : measure_theory.measure β)
(hs : measurable_set s) :
⇑(⇑(measure_theory.measure.comapₗ f) μ) s = ⇑(measure_theory.measure.comap f μ) s
theorem
measure_theory.measure.measure_image_eq_zero_of_comap_eq_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(μ : measure_theory.measure β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)
{s : set α}
(hs : ⇑(measure_theory.measure.comap f μ) s = 0) :
theorem
measure_theory.measure.ae_eq_image_of_ae_eq_comap
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(μ : measure_theory.measure β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)
{s t : set α}
(hst : s =ᵐ[measure_theory.measure.comap f μ] t) :
theorem
measure_theory.measure.null_measurable_set.image
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(μ : measure_theory.measure β)
(hfi : function.injective f)
(hf : ∀ (s : set α), measurable_set s → measure_theory.null_measurable_set (f '' s) μ)
{s : set α}
(hs : measure_theory.null_measurable_set s (measure_theory.measure.comap f μ)) :
measure_theory.null_measurable_set (f '' s) μ
theorem
measure_theory.measure.comap_preimage
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{mβ : measurable_space β}
(f : α → β)
(μ : measure_theory.measure β)
{s : set β}
(hf : function.injective f)
(hf' : measurable f)
(h : ∀ (t : set α), measurable_set t → measure_theory.null_measurable_set (f '' t) μ)
(hs : measurable_set s) :
Subtype of a measure space #
theorem
measure_theory.measure.measurable_set.null_measurable_set_subtype_coe
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{t : set ↥s}
(hs : measure_theory.null_measurable_set s μ)
(ht : measurable_set t) :
theorem
measure_theory.measure.null_measurable_set.subtype_coe
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{t : set ↥s}
(hs : measure_theory.null_measurable_set s μ)
(ht : measure_theory.null_measurable_set t (measure_theory.measure.comap subtype.val μ)) :
theorem
measure_theory.measure.measure_subtype_coe_le_comap
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measure_theory.null_measurable_set s μ)
(t : set ↥s) :
⇑μ (coe '' t) ≤ ⇑(measure_theory.measure.comap subtype.val μ) t
theorem
measure_theory.measure.measure_subtype_coe_eq_zero_of_comap_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measure_theory.null_measurable_set s μ)
{t : set ↥s}
(ht : ⇑(measure_theory.measure.comap subtype.val μ) t = 0) :
@[protected, instance]
noncomputable
def
measure_theory.measure.subtype.measure_space
{α : Type u_1}
[measure_theory.measure_space α]
{p : α → Prop} :
Equations
- measure_theory.measure.subtype.measure_space = {to_measurable_space := {measurable_set' := subtype.measurable_space.measurable_set', measurable_set_empty := _, measurable_set_compl := _, measurable_set_Union := _}, volume := measure_theory.measure.comap subtype.val measure_theory.measure_space.volume}
theorem
measure_theory.measure.volume_subtype_coe_le_volume
{α : Type u_1}
{s : set α}
[measure_theory.measure_space α]
(hs : measure_theory.null_measurable_set s measure_theory.measure_space.volume)
(t : set ↥s) :
theorem
measure_theory.measure.volume_subtype_coe_eq_zero_of_volume_eq_zero
{α : Type u_1}
{s : set α}
[measure_theory.measure_space α]
(hs : measure_theory.null_measurable_set s measure_theory.measure_space.volume)
{t : set ↥s}
(ht : ⇑measure_theory.measure_space.volume t = 0) :
Restricting a measure #
noncomputable
def
measure_theory.measure.restrictₗ
{α : Type u_1}
{m0 : measurable_space α}
(s : set α) :
Restrict a measure μ to a set s as an ℝ≥0∞-linear map.
noncomputable
def
measure_theory.measure.restrict
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(s : set α) :
Restrict a measure μ to a set s.
Equations
- μ.restrict s = ⇑(measure_theory.measure.restrictₗ s) μ
Instances for measure_theory.measure.restrict
- measure_theory.restrict.is_finite_measure
- measure_theory.is_finite_measure_restrict
- measure_theory.measure.restrict.has_no_atoms
- measure_theory.restrict.sigma_finite
- real.is_finite_measure_restrict_Icc
- real.is_finite_measure_restrict_Ico
- real.is_finite_measure_restrict_Ioc
- real.is_finite_measure_restrict_Ioo
- measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict
- measure_theory.measure.restrict.measure_theory.is_finite_measure
@[simp]
theorem
measure_theory.measure.restrictₗ_apply
{α : Type u_1}
{m0 : measurable_space α}
(s : set α)
(μ : measure_theory.measure α) :
⇑(measure_theory.measure.restrictₗ s) μ = μ.restrict s
theorem
measure_theory.measure.restrict_to_outer_measure_eq_to_outer_measure_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(h : measurable_set s) :
This lemma shows that restrict and to_outer_measure commute. Note that the LHS has a
restrict on measures and the RHS has a restrict on outer measures.
theorem
measure_theory.measure.restrict_apply₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(ht : measure_theory.null_measurable_set t (μ.restrict s)) :
@[simp]
theorem
measure_theory.measure.restrict_apply
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(ht : measurable_set t) :
If t is a measurable set, then the measure of t with respect to the restriction of
the measure to s equals the outer measure of t ∩ s. An alternate version requiring that s
be measurable instead of t exists as measure.restrict_apply'.
theorem
measure_theory.measure.restrict_mono'
{α : Type u_1}
{m0 : measurable_space α}
⦃s s' : set α⦄
⦃μ ν : measure_theory.measure α⦄
(hs : s ≤ᵐ[μ] s')
(hμν : μ ≤ ν) :
Restriction of a measure to a subset is monotone both in set and in measure.
theorem
measure_theory.measure.restrict_mono
{α : Type u_1}
{m0 : measurable_space α}
⦃s s' : set α⦄
(hs : s ⊆ s')
⦃μ ν : measure_theory.measure α⦄
(hμν : μ ≤ ν) :
Restriction of a measure to a subset is monotone both in set and in measure.
theorem
measure_theory.measure.restrict_mono_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : s ≤ᵐ[μ] t) :
theorem
measure_theory.measure.restrict_congr_set
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : s =ᵐ[μ] t) :
@[simp]
theorem
measure_theory.measure.restrict_apply'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s) :
If s is a measurable set, then the outer measure of t with respect to the restriction of
the measure to s equals the outer measure of t ∩ s. This is an alternate version of
measure.restrict_apply, requiring that s is measurable instead of t.
theorem
measure_theory.measure.restrict_apply₀'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measure_theory.null_measurable_set s μ) :
theorem
measure_theory.measure.restrict_le_self
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
theorem
measure_theory.measure.restrict_eq_self
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
{s t : set α}
(h : s ⊆ t) :
@[simp]
theorem
measure_theory.measure.restrict_apply_self
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(s : set α) :
theorem
measure_theory.measure.restrict_apply_univ
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s : set α) :
theorem
measure_theory.measure.le_restrict_apply
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s t : set α) :
theorem
measure_theory.measure.restrict_apply_superset
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : s ⊆ t) :
@[simp]
theorem
measure_theory.measure.restrict_add
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : measure_theory.measure α)
(s : set α) :
@[simp]
@[simp]
theorem
measure_theory.measure.restrict_smul
{α : Type u_1}
{m0 : measurable_space α}
(c : ennreal)
(μ : measure_theory.measure α)
(s : set α) :
theorem
measure_theory.measure.restrict_restrict₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measure_theory.null_measurable_set s (μ.restrict t)) :
@[simp]
theorem
measure_theory.measure.restrict_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.restrict_restrict_of_subset
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : s ⊆ t) :
theorem
measure_theory.measure.restrict_restrict₀'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(ht : measure_theory.null_measurable_set t μ) :
theorem
measure_theory.measure.restrict_restrict'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_comm
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.restrict_apply_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(ht : measurable_set t) :
theorem
measure_theory.measure.measure_inter_eq_zero_of_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : ⇑(μ.restrict s) t = 0) :
theorem
measure_theory.measure.restrict_apply_eq_zero'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(hs : measurable_set s) :
@[simp]
theorem
measure_theory.measure.restrict_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
theorem
measure_theory.measure.restrict_zero_set
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(h : ⇑μ s = 0) :
@[simp]
theorem
measure_theory.measure.restrict_empty
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.measure.restrict_univ
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.measure.restrict_inter_add_diff₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measure_theory.null_measurable_set t μ) :
theorem
measure_theory.measure.restrict_inter_add_diff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_union_add_inter₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measure_theory.null_measurable_set t μ) :
theorem
measure_theory.measure.restrict_union_add_inter
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{t : set α}
(s : set α)
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_union_add_inter'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(t : set α) :
theorem
measure_theory.measure.restrict_union₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : measure_theory.ae_disjoint μ s t)
(ht : measure_theory.null_measurable_set t μ) :
theorem
measure_theory.measure.restrict_union
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : disjoint s t)
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_union'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
(h : disjoint s t)
(hs : measurable_set s) :
@[simp]
theorem
measure_theory.measure.restrict_add_restrict_compl
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
@[simp]
theorem
measure_theory.measure.restrict_compl_add_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.restrict_union_le
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s s' : set α) :
theorem
measure_theory.measure.restrict_Union_apply_ae
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : pairwise (measure_theory.ae_disjoint μ on s))
(hm : ∀ (i : ι), measure_theory.null_measurable_set (s i) μ)
{t : set α}
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_Union_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : pairwise (disjoint on s))
(hm : ∀ (i : ι), measurable_set (s i))
{t : set α}
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_Union_apply_eq_supr
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : directed has_subset.subset s)
{t : set α}
(ht : measurable_set t) :
theorem
measure_theory.measure.restrict_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : measurable f)
{s : set β}
(hs : measurable_set s) :
(measure_theory.measure.map f μ).restrict s = measure_theory.measure.map f (μ.restrict (f ⁻¹' s))
The restriction of the pushforward measure is the pushforward of the restriction. For a version
assuming only ae_measurable, see restrict_map_of_ae_measurable.
theorem
measure_theory.measure.restrict_to_measurable
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(h : ⇑μ s ≠ ⊤) :
μ.restrict (measure_theory.to_measurable μ s) = μ.restrict s
theorem
measure_theory.measure.restrict_eq_self_of_ae_mem
{α : Type u_1}
{m0 : measurable_space α}
⦃s : set α⦄
⦃μ : measure_theory.measure α⦄
(hs : ∀ᵐ (x : α) ∂μ, x ∈ s) :
theorem
measure_theory.measure.restrict_congr_meas
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.restrict_congr_mono
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s t : set α}
(hs : s ⊆ t)
(h : μ.restrict t = ν.restrict t) :
theorem
measure_theory.measure.restrict_union_congr
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s t : set α} :
If two measures agree on all measurable subsets of s and t, then they agree on all
measurable subsets of s ∪ t.
theorem
measure_theory.measure.restrict_finset_bUnion_congr
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s : finset ι}
{t : ι → set α} :
theorem
measure_theory.measure.restrict_bUnion_congr
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s : set ι}
{t : ι → set α}
(hc : s.countable) :
theorem
measure_theory.measure.restrict_Inf_eq_Inf_restrict
{α : Type u_1}
{t : set α}
{m0 : measurable_space α}
{m : set (measure_theory.measure α)}
(hm : m.nonempty)
(ht : measurable_set t) :
(has_Inf.Inf m).restrict t = has_Inf.Inf ((λ (μ : measure_theory.measure α), μ.restrict t) '' m)
This lemma shows that Inf and restrict commute for measures.
Extensionality results #
theorem
measure_theory.measure.ext_iff_of_Union_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hs : (⋃ (i : ι), s i) = set.univ) :
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using Union).
theorem
measure_theory.measure.ext_of_Union_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hs : (⋃ (i : ι), s i) = set.univ) :
Alias of the reverse direction of measure_theory.measure.ext_iff_of_Union_eq_univ.
theorem
measure_theory.measure.ext_iff_of_bUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S : set ι}
{s : ι → set α}
(hc : S.countable)
(hs : (⋃ (i : ι) (H : i ∈ S), s i) = set.univ) :
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using bUnion).
theorem
measure_theory.measure.ext_of_bUnion_eq_univ
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S : set ι}
{s : ι → set α}
(hc : S.countable)
(hs : (⋃ (i : ι) (H : i ∈ S), s i) = set.univ) :
Alias of the reverse direction of measure_theory.measure.ext_iff_of_bUnion_eq_univ.
theorem
measure_theory.measure.ext_iff_of_sUnion_eq_univ
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S : set (set α)}
(hc : S.countable)
(hs : ⋃₀ S = set.univ) :
Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using sUnion).
theorem
measure_theory.measure.ext_of_sUnion_eq_univ
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S : set (set α)}
(hc : S.countable)
(hs : ⋃₀ S = set.univ) :
Alias of the reverse direction of measure_theory.measure.ext_iff_of_sUnion_eq_univ.
theorem
measure_theory.measure.ext_of_generate_from_of_cover
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S T : set (set α)}
(h_gen : m0 = measurable_space.generate_from S)
(hc : T.countable)
(h_inter : is_pi_system S)
(hU : ⋃₀ T = set.univ)
(htop : ∀ (t : set α), t ∈ T → ⇑μ t ≠ ⊤)
(ST_eq : ∀ (t : set α), t ∈ T → ∀ (s : set α), s ∈ S → ⇑μ (s ∩ t) = ⇑ν (s ∩ t))
(T_eq : ∀ (t : set α), t ∈ T → ⇑μ t = ⇑ν t) :
μ = ν
theorem
measure_theory.measure.ext_of_generate_from_of_cover_subset
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{S T : set (set α)}
(h_gen : m0 = measurable_space.generate_from S)
(h_inter : is_pi_system S)
(h_sub : T ⊆ S)
(hc : T.countable)
(hU : ⋃₀ T = set.univ)
(htop : ∀ (s : set α), s ∈ T → ⇑μ s ≠ ⊤)
(h_eq : ∀ (s : set α), s ∈ S → ⇑μ s = ⇑ν s) :
μ = ν
Two measures are equal if they are equal on the π-system generating the σ-algebra,
and they are both finite on a increasing spanning sequence of sets in the π-system.
This lemma is formulated using sUnion.
theorem
measure_theory.measure.ext_of_generate_from_of_Union
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(C : set (set α))
(B : ℕ → set α)
(hA : m0 = measurable_space.generate_from C)
(hC : is_pi_system C)
(h1B : (⋃ (i : ℕ), B i) = set.univ)
(h2B : ∀ (i : ℕ), B i ∈ C)
(hμB : ∀ (i : ℕ), ⇑μ (B i) ≠ ⊤)
(h_eq : ∀ (s : set α), s ∈ C → ⇑μ s = ⇑ν s) :
μ = ν
Two measures are equal if they are equal on the π-system generating the σ-algebra,
and they are both finite on a increasing spanning sequence of sets in the π-system.
This lemma is formulated using Union.
finite_spanning_sets_in.ext is a reformulation of this lemma.
The dirac measure.
Equations
Instances for measure_theory.measure.dirac
@[protected, instance]
Equations
theorem
measure_theory.measure.le_dirac_apply
{α : Type u_1}
{s : set α}
[measurable_space α]
{a : α} :
s.indicator 1 a ≤ ⇑(measure_theory.measure.dirac a) s
@[simp]
theorem
measure_theory.measure.dirac_apply'
{α : Type u_1}
{s : set α}
[measurable_space α]
(a : α)
(hs : measurable_set s) :
⇑(measure_theory.measure.dirac a) s = s.indicator 1 a
@[simp]
theorem
measure_theory.measure.dirac_apply_of_mem
{α : Type u_1}
{s : set α}
[measurable_space α]
{a : α}
(h : a ∈ s) :
⇑(measure_theory.measure.dirac a) s = 1
@[simp]
theorem
measure_theory.measure.dirac_apply
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(a : α)
(s : set α) :
⇑(measure_theory.measure.dirac a) s = s.indicator 1 a
theorem
measure_theory.measure.map_dirac
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
{f : α → β}
(hf : measurable f)
(a : α) :
@[simp]
theorem
measure_theory.measure.restrict_singleton
{α : Type u_1}
[measurable_space α]
(μ : measure_theory.measure α)
(a : α) :
μ.restrict {a} = ⇑μ {a} • measure_theory.measure.dirac a
noncomputable
def
measure_theory.measure.sum
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(f : ι → measure_theory.measure α) :
Sum of an indexed family of measures.
Equations
- measure_theory.measure.sum f = (measure_theory.outer_measure.sum (λ (i : ι), (f i).to_outer_measure)).to_measure _
Instances for measure_theory.measure.sum
theorem
measure_theory.measure.le_sum_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(f : ι → measure_theory.measure α)
(s : set α) :
@[simp]
theorem
measure_theory.measure.sum_apply
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(f : ι → measure_theory.measure α)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.le_sum
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(μ : ι → measure_theory.measure α)
(i : ι) :
μ i ≤ measure_theory.measure.sum μ
@[simp]
theorem
measure_theory.measure.sum_apply_eq_zero
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
[countable ι]
{μ : ι → measure_theory.measure α}
{s : set α} :
theorem
measure_theory.measure.sum_apply_eq_zero'
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : ι → measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.sum_comm
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{ι' : Type u_2}
(μ : ι → ι' → measure_theory.measure α) :
measure_theory.measure.sum (λ (n : ι), measure_theory.measure.sum (μ n)) = measure_theory.measure.sum (λ (m : ι'), measure_theory.measure.sum (λ (n : ι), μ n m))
theorem
measure_theory.measure.ae_sum_iff
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
[countable ι]
{μ : ι → measure_theory.measure α}
{p : α → Prop} :
theorem
measure_theory.measure.ae_sum_iff'
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : ι → measure_theory.measure α}
{p : α → Prop}
(h : measurable_set {x : α | p x}) :
@[simp]
theorem
measure_theory.measure.sum_fintype
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
[fintype ι]
(μ : ι → measure_theory.measure α) :
measure_theory.measure.sum μ = finset.univ.sum (λ (i : ι), μ i)
@[simp]
theorem
measure_theory.measure.sum_coe_finset
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(s : finset ι)
(μ : ι → measure_theory.measure α) :
measure_theory.measure.sum (λ (i : ↥s), μ ↑i) = s.sum (λ (i : ι), μ i)
@[simp]
theorem
measure_theory.measure.ae_sum_eq
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
[countable ι]
(μ : ι → measure_theory.measure α) :
@[simp]
theorem
measure_theory.measure.sum_bool
{α : Type u_1}
{m0 : measurable_space α}
(f : bool → measure_theory.measure α) :
@[simp]
theorem
measure_theory.measure.sum_cond
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : measure_theory.measure α) :
measure_theory.measure.sum (λ (b : bool), cond b μ ν) = μ + ν
@[simp]
theorem
measure_theory.measure.restrict_sum
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(μ : ι → measure_theory.measure α)
{s : set α}
(hs : measurable_set s) :
(measure_theory.measure.sum μ).restrict s = measure_theory.measure.sum (λ (i : ι), (μ i).restrict s)
@[simp]
theorem
measure_theory.measure.sum_of_empty
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
[is_empty ι]
(μ : ι → measure_theory.measure α) :
theorem
measure_theory.measure.sum_add_sum_compl
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
(s : set ι)
(μ : ι → measure_theory.measure α) :
measure_theory.measure.sum (λ (i : ↥s), μ ↑i) + measure_theory.measure.sum (λ (i : ↥sᶜ), μ ↑i) = measure_theory.measure.sum μ
theorem
measure_theory.measure.sum_congr
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : ℕ → measure_theory.measure α}
(h : ∀ (n : ℕ), μ n = ν n) :
theorem
measure_theory.measure.sum_add_sum
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : ℕ → measure_theory.measure α) :
measure_theory.measure.sum μ + measure_theory.measure.sum ν = measure_theory.measure.sum (λ (n : ℕ), μ n + ν n)
theorem
measure_theory.measure.map_eq_sum
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
[countable β]
[measurable_singleton_class β]
(μ : measure_theory.measure α)
(f : α → β)
(hf : measurable f) :
measure_theory.measure.map f μ = measure_theory.measure.sum (λ (b : β), ⇑μ (f ⁻¹' {b}) • measure_theory.measure.dirac b)
If f is a map with countable codomain, then μ.map f is a sum of Dirac measures.
@[simp]
theorem
measure_theory.measure.sum_smul_dirac
{α : Type u_1}
{m0 : measurable_space α}
[countable α]
[measurable_singleton_class α]
(μ : measure_theory.measure α) :
measure_theory.measure.sum (λ (a : α), ⇑μ {a} • measure_theory.measure.dirac a) = μ
A measure on a countable type is a sum of Dirac measures.
theorem
measure_theory.measure.tsum_indicator_apply_singleton
{α : Type u_1}
{m0 : measurable_space α}
[countable α]
[measurable_singleton_class α]
(μ : measure_theory.measure α)
(s : set α)
(hs : measurable_set s) :
Given that α is a countable, measurable space with all singleton sets measurable,
write the measure of a set s as the sum of the measure of {x} for all x ∈ s.
theorem
measure_theory.measure.restrict_Union_ae
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : pairwise (measure_theory.ae_disjoint μ on s))
(hm : ∀ (i : ι), measure_theory.null_measurable_set (s i) μ) :
theorem
measure_theory.measure.restrict_Union
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α}
(hd : pairwise (disjoint on s))
(hm : ∀ (i : ι), measurable_set (s i)) :
theorem
measure_theory.measure.restrict_Union_le
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[countable ι]
{s : ι → set α} :
Counting measure on any measurable space.
Instances for measure_theory.measure.count
theorem
measure_theory.measure.count_apply
{α : Type u_1}
{s : set α}
[measurable_space α]
(hs : measurable_set s) :
@[simp]
@[simp]
theorem
measure_theory.measure.count_apply_finset'
{α : Type u_1}
[measurable_space α]
{s : finset α}
(s_mble : measurable_set ↑s) :
@[simp]
theorem
measure_theory.measure.count_apply_finset
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(s : finset α) :
theorem
measure_theory.measure.count_apply_finite'
{α : Type u_1}
[measurable_space α]
{s : set α}
(s_fin : s.finite)
(s_mble : measurable_set s) :
theorem
measure_theory.measure.count_apply_finite
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(s : set α)
(hs : s.finite) :
theorem
measure_theory.measure.count_apply_infinite
{α : Type u_1}
{s : set α}
[measurable_space α]
(hs : s.infinite) :
count measure evaluates to infinity at infinite sets.
@[simp]
theorem
measure_theory.measure.count_apply_eq_top'
{α : Type u_1}
{s : set α}
[measurable_space α]
(s_mble : measurable_set s) :
@[simp]
theorem
measure_theory.measure.count_apply_eq_top
{α : Type u_1}
{s : set α}
[measurable_space α]
[measurable_singleton_class α] :
@[simp]
theorem
measure_theory.measure.count_apply_lt_top'
{α : Type u_1}
{s : set α}
[measurable_space α]
(s_mble : measurable_set s) :
@[simp]
theorem
measure_theory.measure.count_apply_lt_top
{α : Type u_1}
{s : set α}
[measurable_space α]
[measurable_singleton_class α] :
theorem
measure_theory.measure.empty_of_count_eq_zero'
{α : Type u_1}
{s : set α}
[measurable_space α]
(s_mble : measurable_set s)
(hsc : ⇑measure_theory.measure.count s = 0) :
theorem
measure_theory.measure.empty_of_count_eq_zero
{α : Type u_1}
{s : set α}
[measurable_space α]
[measurable_singleton_class α]
(hsc : ⇑measure_theory.measure.count s = 0) :
@[simp]
theorem
measure_theory.measure.count_eq_zero_iff'
{α : Type u_1}
{s : set α}
[measurable_space α]
(s_mble : measurable_set s) :
@[simp]
theorem
measure_theory.measure.count_eq_zero_iff
{α : Type u_1}
{s : set α}
[measurable_space α]
[measurable_singleton_class α] :
theorem
measure_theory.measure.count_ne_zero'
{α : Type u_1}
{s : set α}
[measurable_space α]
(hs' : s.nonempty)
(s_mble : measurable_set s) :
theorem
measure_theory.measure.count_ne_zero
{α : Type u_1}
{s : set α}
[measurable_space α]
[measurable_singleton_class α]
(hs' : s.nonempty) :
@[simp]
theorem
measure_theory.measure.count_singleton'
{α : Type u_1}
[measurable_space α]
{a : α}
(ha : measurable_set {a}) :
@[simp]
theorem
measure_theory.measure.count_singleton
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(a : α) :
theorem
measure_theory.measure.count_injective_image'
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
{f : β → α}
(hf : function.injective f)
{s : set β}
(s_mble : measurable_set s)
(fs_mble : measurable_set (f '' s)) :
theorem
measure_theory.measure.count_injective_image
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
[measurable_singleton_class α]
[measurable_singleton_class β]
{f : β → α}
(hf : function.injective f)
(s : set β) :
Absolute continuity #
def
measure_theory.measure.absolutely_continuous
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : measure_theory.measure α) :
Prop
We say that μ is absolutely continuous with respect to ν, or that μ is dominated by ν,
if ν(A) = 0 implies that μ(A) = 0.
Instances for measure_theory.measure.absolutely_continuous
theorem
measure_theory.measure.absolutely_continuous_of_le
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ ≤ ν) :
theorem
has_le.le.absolutely_continuous
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ ≤ ν) :
Alias of measure_theory.measure.absolutely_continuous_of_le.
theorem
measure_theory.measure.absolutely_continuous_of_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ = ν) :
theorem
eq.absolutely_continuous
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ = ν) :
Alias of measure_theory.measure.absolutely_continuous_of_eq.
theorem
measure_theory.measure.absolutely_continuous.mk
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : ∀ ⦃s : set α⦄, measurable_set s → ⇑ν s = 0 → ⇑μ s = 0) :
@[protected, refl]
theorem
measure_theory.measure.absolutely_continuous.refl
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
@[protected]
theorem
measure_theory.measure.absolutely_continuous.rfl
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[protected, instance]
@[protected, trans]
theorem
measure_theory.measure.absolutely_continuous.trans
{α : Type u_1}
{m0 : measurable_space α}
{μ₁ μ₂ μ₃ : measure_theory.measure α}
(h1 : μ₁.absolutely_continuous μ₂)
(h2 : μ₂.absolutely_continuous μ₃) :
μ₁.absolutely_continuous μ₃
@[protected]
theorem
measure_theory.measure.absolutely_continuous.map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ ν : measure_theory.measure α}
(h : μ.absolutely_continuous ν)
{f : α → β}
(hf : measurable f) :
@[protected]
theorem
measure_theory.measure.absolutely_continuous.smul
{α : Type u_1}
{R : Type u_6}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
[monoid R]
[distrib_mul_action R ennreal]
[is_scalar_tower R ennreal ennreal]
(h : μ.absolutely_continuous ν)
(c : R) :
(c • μ).absolutely_continuous ν
theorem
measure_theory.measure.absolutely_continuous_of_le_smul
{α : Type u_1}
{m0 : measurable_space α}
{μ μ' : measure_theory.measure α}
{c : ennreal}
(hμ'_le : μ' ≤ c • μ) :
theorem
measure_theory.measure.ae_le_iff_absolutely_continuous
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
μ.ae ≤ ν.ae ↔ μ.absolutely_continuous ν
theorem
has_le.le.absolutely_continuous_of_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
μ.ae ≤ ν.ae → μ.absolutely_continuous ν
Alias of the forward direction of measure_theory.measure.ae_le_iff_absolutely_continuous.
theorem
measure_theory.measure.absolutely_continuous.ae_le
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
μ.absolutely_continuous ν → μ.ae ≤ ν.ae
Alias of the reverse direction of measure_theory.measure.ae_le_iff_absolutely_continuous.
theorem
measure_theory.measure.ae_mono'
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α} :
μ.absolutely_continuous ν → μ.ae ≤ ν.ae
Alias of measure_theory.measure.absolutely_continuous.ae_le.
theorem
measure_theory.measure.absolutely_continuous.ae_eq
{α : Type u_1}
{δ : Type u_4}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ.absolutely_continuous ν)
{f g : α → δ}
(h' : f =ᵐ[ν] g) :
Quasi measure preserving maps (a.k.a. non-singular maps) #
structure
measure_theory.measure.quasi_measure_preserving
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{m0 : measurable_space α}
(f : α → β)
(μa : measure_theory.measure α . "volume_tac")
(μb : measure_theory.measure β . "volume_tac") :
Prop
- measurable : measurable f
- absolutely_continuous : (measure_theory.measure.map f μa).absolutely_continuous μb
A map f : α → β is said to be quasi measure preserving (a.k.a. non-singular) w.r.t. measures
μa and μb if it is measurable and μb s = 0 implies μa (f ⁻¹' s) = 0.
@[protected]
theorem
measure_theory.measure.quasi_measure_preserving.id
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
@[protected]
theorem
measurable.quasi_measure_preserving
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{f : α → β}
{m0 : measurable_space α}
(hf : measurable f)
(μ : measure_theory.measure α) :
theorem
measure_theory.measure.quasi_measure_preserving.mono_left
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa μa' : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb)
(ha : μa'.absolutely_continuous μa) :
theorem
measure_theory.measure.quasi_measure_preserving.mono_right
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb μb' : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb)
(ha : μb.absolutely_continuous μb') :
theorem
measure_theory.measure.quasi_measure_preserving.mono
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa μa' : measure_theory.measure α}
{μb μb' : measure_theory.measure β}
{f : α → β}
(ha : μa'.absolutely_continuous μa)
(hb : μb.absolutely_continuous μb')
(h : measure_theory.measure.quasi_measure_preserving f μa μb) :
@[protected]
theorem
measure_theory.measure.quasi_measure_preserving.comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m0 : measurable_space α}
[measurable_space β]
[measurable_space γ]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{μc : measure_theory.measure γ}
{g : β → γ}
{f : α → β}
(hg : measure_theory.measure.quasi_measure_preserving g μb μc)
(hf : measure_theory.measure.quasi_measure_preserving f μa μb) :
measure_theory.measure.quasi_measure_preserving (g ∘ f) μa μc
@[protected]
theorem
measure_theory.measure.quasi_measure_preserving.iterate
{α : Type u_1}
{m0 : measurable_space α}
{μa : measure_theory.measure α}
{f : α → α}
(hf : measure_theory.measure.quasi_measure_preserving f μa μa)
(n : ℕ) :
@[protected]
theorem
measure_theory.measure.quasi_measure_preserving.ae_measurable
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(hf : measure_theory.measure.quasi_measure_preserving f μa μb) :
ae_measurable f μa
theorem
measure_theory.measure.quasi_measure_preserving.ae_map_le
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb) :
(measure_theory.measure.map f μa).ae ≤ μb.ae
theorem
measure_theory.measure.quasi_measure_preserving.tendsto_ae
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb) :
filter.tendsto f μa.ae μb.ae
theorem
measure_theory.measure.quasi_measure_preserving.ae
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb)
{p : β → Prop}
(hg : ∀ᵐ (x : β) ∂μb, p x) :
theorem
measure_theory.measure.quasi_measure_preserving.ae_eq
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb)
{g₁ g₂ : β → δ}
(hg : g₁ =ᵐ[μb] g₂) :
theorem
measure_theory.measure.quasi_measure_preserving.preimage_null
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
(h : measure_theory.measure.quasi_measure_preserving f μa μb)
{s : set β}
(hs : ⇑μb s = 0) :
theorem
measure_theory.measure.quasi_measure_preserving.preimage_mono_ae
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
{s t : set β}
(hf : measure_theory.measure.quasi_measure_preserving f μa μb)
(h : s ≤ᵐ[μb] t) :
theorem
measure_theory.measure.quasi_measure_preserving.preimage_ae_eq
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{f : α → β}
{s t : set β}
(hf : measure_theory.measure.quasi_measure_preserving f μa μb)
(h : s =ᵐ[μb] t) :
theorem
measure_theory.measure.quasi_measure_preserving.preimage_iterate_ae_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f : α → α}
(hf : measure_theory.measure.quasi_measure_preserving f μ μ)
(k : ℕ)
(hs : f ⁻¹' s =ᵐ[μ] s) :
theorem
measure_theory.measure.quasi_measure_preserving.image_zpow_ae_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{e : α ≃ α}
(he : measure_theory.measure.quasi_measure_preserving ⇑e μ μ)
(he' : measure_theory.measure.quasi_measure_preserving ⇑(e.symm) μ μ)
(k : ℤ)
(hs : ⇑e '' s =ᵐ[μ] s) :
theorem
measure_theory.measure.quasi_measure_preserving.limsup_preimage_iterate_ae_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f : α → α}
(hf : measure_theory.measure.quasi_measure_preserving f μ μ)
(hs : f ⁻¹' s =ᵐ[μ] s) :
filter.limsup (λ (n : ℕ), set.preimage f^[n] s) filter.at_top =ᵐ[μ] s
theorem
measure_theory.measure.quasi_measure_preserving.liminf_preimage_iterate_ae_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f : α → α}
(hf : measure_theory.measure.quasi_measure_preserving f μ μ)
(hs : f ⁻¹' s =ᵐ[μ] s) :
filter.liminf (λ (n : ℕ), set.preimage f^[n] s) filter.at_top =ᵐ[μ] s
theorem
measure_theory.measure.quasi_measure_preserving.exists_preimage_eq_of_preimage_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f : α → α}
(h : measure_theory.measure.quasi_measure_preserving f μ μ)
(hs : measurable_set s)
(hs' : f ⁻¹' s =ᵐ[μ] s) :
By replacing a measurable set that is almost invariant with the limsup of its preimages, we
obtain a measurable set that is almost equal and strictly invariant.
(The liminf would work just as well.)
theorem
measure_theory.measure.quasi_measure_preserving.smul_ae_eq_of_ae_eq
{G : Type u_1}
{α : Type u_2}
[group G]
[mul_action G α]
[measurable_space α]
{s t : set α}
{μ : measure_theory.measure α}
(g : G)
(h_qmp : measure_theory.measure.quasi_measure_preserving (has_smul.smul g⁻¹) μ μ)
(h_ae_eq : s =ᵐ[μ] t) :
theorem
measure_theory.measure.quasi_measure_preserving.vadd_ae_eq_of_ae_eq
{G : Type u_1}
{α : Type u_2}
[add_group G]
[add_action G α]
[measurable_space α]
{s t : set α}
{μ : measure_theory.measure α}
(g : G)
(h_qmp : measure_theory.measure.quasi_measure_preserving (has_vadd.vadd (-g)) μ μ)
(h_ae_eq : s =ᵐ[μ] t) :
theorem
measure_theory.measure.pairwise_ae_disjoint_of_ae_disjoint_forall_ne_one
{G : Type u_1}
{α : Type u_2}
[group G]
[mul_action G α]
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
(h_ae_disjoint : ∀ (g : G), g ≠ 1 → measure_theory.ae_disjoint μ (g • s) s)
(h_qmp : ∀ (g : G), measure_theory.measure.quasi_measure_preserving (has_smul.smul g) μ μ) :
pairwise (measure_theory.ae_disjoint μ on λ (g : G), g • s)
theorem
measure_theory.measure.pairwise_ae_disjoint_of_ae_disjoint_forall_ne_zero
{G : Type u_1}
{α : Type u_2}
[add_group G]
[add_action G α]
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
(h_ae_disjoint : ∀ (g : G), g ≠ 0 → measure_theory.ae_disjoint μ (g +ᵥ s) s)
(h_qmp : ∀ (g : G), measure_theory.measure.quasi_measure_preserving (has_vadd.vadd g) μ μ) :
pairwise (measure_theory.ae_disjoint μ on λ (g : G), g +ᵥ s)
The cofinite filter #
def
measure_theory.measure.cofinite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
filter α
The filter of sets s such that sᶜ has finite measure.
theorem
measure_theory.measure.mem_cofinite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
theorem
measure_theory.measure.compl_mem_cofinite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
theorem
measure_theory.measure.eventually_cofinite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{p : α → Prop} :
theorem
measure_theory.null_measurable_set.preimage
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → β}
{t : set β}
(ht : measure_theory.null_measurable_set t ν)
(hf : measure_theory.measure.quasi_measure_preserving f μ ν) :
The preimage of a null measurable set under a (quasi) measure preserving map is a null measurable set.
theorem
measure_theory.null_measurable_set.mono_ac
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s : set α}
(h : measure_theory.null_measurable_set s μ)
(hle : ν.absolutely_continuous μ) :
theorem
measure_theory.null_measurable_set.mono
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{s : set α}
(h : measure_theory.null_measurable_set s μ)
(hle : ν ≤ μ) :
theorem
measure_theory.ae_disjoint.preimage
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → β}
{s t : set β}
(ht : measure_theory.ae_disjoint ν s t)
(hf : measure_theory.measure.quasi_measure_preserving f μ ν) :
measure_theory.ae_disjoint μ (f ⁻¹' s) (f ⁻¹' t)
@[simp]
theorem
measure_theory.ae_eq_bot
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[simp]
theorem
measure_theory.ae_ne_bot
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.ae_mono
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : μ ≤ ν) :
theorem
measure_theory.mem_ae_map_iff
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{s : set β}
(hs : measurable_set s) :
theorem
measure_theory.mem_ae_of_mem_ae_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{s : set β}
(hs : s ∈ (measure_theory.measure.map f μ).ae) :
theorem
measure_theory.ae_map_iff
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{p : β → Prop}
(hp : measurable_set {x : β | p x}) :
theorem
measure_theory.ae_of_ae_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
(hf : ae_measurable f μ)
{p : β → Prop}
(h : ∀ᵐ (y : β) ∂measure_theory.measure.map f μ, p y) :
theorem
measure_theory.ae_map_mem_range
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{m0 : measurable_space α}
(f : α → β)
(hf : measurable_set (set.range f))
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.ae_restrict_union_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s t : set α) :
theorem
measure_theory.ae_restrict_bUnion_iff
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s : ι → set α)
{t : set ι}
(ht : t.countable)
(p : α → Prop) :
@[simp]
theorem
measure_theory.ae_restrict_bUnion_finset_iff
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s : ι → set α)
(t : finset ι)
(p : α → Prop) :
theorem
measure_theory.ae_eq_restrict_bUnion_iff
{α : Type u_1}
{δ : Type u_4}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(s : ι → set α)
{t : set ι}
(ht : t.countable)
(f g : α → δ) :
theorem
measure_theory.ae_restrict_uIoc_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[linear_order α]
(a b : α) :
theorem
measure_theory.ae_restrict_uIoc_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[linear_order α]
{a b : α}
{P : α → Prop} :
See also measure_theory.ae_uIoc_iff.
theorem
measure_theory.ae_restrict_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{p : α → Prop}
(hp : measurable_set {x : α | p x}) :
theorem
measure_theory.ae_restrict_iff'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{p : α → Prop}
(hs : measurable_set s) :
theorem
filter.eventually_eq.restrict
{α : Type u_1}
{δ : Type u_4}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → δ}
{s : set α}
(hfg : f =ᵐ[μ] g) :
theorem
measure_theory.ae_restrict_mem
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.ae_restrict_mem₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measure_theory.null_measurable_set s μ) :
theorem
measure_theory.ae_restrict_of_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{p : α → Prop}
(h : ∀ᵐ (x : α) ∂μ, p x) :
theorem
measure_theory.ae_restrict_iff'₀
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{p : α → Prop}
(hs : measure_theory.null_measurable_set s μ) :
theorem
measure_theory.mem_map_restrict_ae_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{β : Type u_2}
{s : set α}
{t : set β}
{f : α → β}
(hs : measurable_set s) :
theorem
measure_theory.ae_smul_measure
{α : Type u_1}
{R : Type u_6}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{p : α → Prop}
[monoid R]
[distrib_mul_action R ennreal]
[is_scalar_tower R ennreal ennreal]
(h : ∀ᵐ (x : α) ∂μ, p x)
(c : R) :
theorem
measure_theory.ae_add_measure_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{p : α → Prop}
{ν : measure_theory.measure α} :
theorem
measure_theory.ae_eq_comp'
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → β}
{g g' : β → δ}
(hf : ae_measurable f μ)
(h : g =ᵐ[ν] g')
(h2 : (measure_theory.measure.map f μ).absolutely_continuous ν) :
theorem
measure_theory.measure.quasi_measure_preserving.ae_eq_comp
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{f : α → β}
{g g' : β → δ}
(hf : measure_theory.measure.quasi_measure_preserving f μ ν)
(h : g =ᵐ[ν] g') :
theorem
measure_theory.ae_eq_comp
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
{f : α → β}
{g g' : β → δ}
(hf : ae_measurable f μ)
(h : g =ᵐ[measure_theory.measure.map f μ] g') :
theorem
measure_theory.sub_ae_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{β : Type u_2}
[add_group β]
(f g : α → β) :
theorem
measure_theory.le_ae_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
@[simp]
theorem
measure_theory.ae_restrict_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
@[simp]
theorem
measure_theory.ae_restrict_eq_bot
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
@[simp]
theorem
measure_theory.ae_restrict_ne_bot
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
theorem
measure_theory.self_mem_ae_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.ae_restrict_of_ae_eq_of_ae_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : α → Prop}
(hst : s =ᵐ[μ] t)
{p : α → Prop} :
If two measurable sets are ae_eq then any proposition that is almost everywhere true on one is almost everywhere true on the other
theorem
measure_theory.ae_restrict_congr_set
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : α → Prop}
(hst : s =ᵐ[μ] t)
{p : α → Prop} :
If two measurable sets are ae_eq then any proposition that is almost everywhere true on one is almost everywhere true on the other
theorem
measure_theory.measure_set_of_frequently_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{p : ℕ → α → Prop}
(hp : ∑' (i : ℕ), ⇑μ {x : α | p i x} ≠ ⊤) :
A version of the Borel-Cantelli lemma: if pᵢ is a sequence of predicates such that
∑ μ {x | pᵢ x} is finite, then the measure of x such that pᵢ x holds frequently as i → ∞ (or
equivalently, pᵢ x holds for infinitely many i) is equal to zero.
theorem
measure_theory.ae_eventually_not_mem
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : ℕ → set α}
(hs : ∑' (i : ℕ), ⇑μ (s i) ≠ ⊤) :
A version of the Borel-Cantelli lemma: if sᵢ is a sequence of sets such that
∑ μ sᵢ exists, then for almost all x, x does not belong to almost all sᵢ.
theorem
measure_theory.Iio_ae_eq_Iic'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.Ioi_ae_eq_Ici'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.Ioo_ae_eq_Ioc'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.Ioc_ae_eq_Icc'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.Ioo_ae_eq_Ico'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0) :
theorem
measure_theory.Ioo_ae_eq_Icc'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0)
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.Ico_ae_eq_Icc'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.Ico_ae_eq_Ioc'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[partial_order α]
{a b : α}
(ha : ⇑μ {a} = 0)
(hb : ⇑μ {b} = 0) :
theorem
measure_theory.mem_ae_dirac_iff
{α : Type u_1}
{s : set α}
[measurable_space α]
{a : α}
(hs : measurable_set s) :
s ∈ (measure_theory.measure.dirac a).ae ↔ a ∈ s
theorem
measure_theory.ae_dirac_iff
{α : Type u_1}
[measurable_space α]
{a : α}
{p : α → Prop}
(hp : measurable_set {x : α | p x}) :
(∀ᵐ (x : α) ∂measure_theory.measure.dirac a, p x) ↔ p a
@[simp]
theorem
measure_theory.ae_dirac_eq
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(a : α) :
theorem
measure_theory.ae_eq_dirac'
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
[measurable_space α]
[measurable_singleton_class β]
{a : α}
{f : α → β}
(hf : measurable f) :
f =ᵐ[measure_theory.measure.dirac a] function.const α (f a)
theorem
measure_theory.ae_eq_dirac
{α : Type u_1}
{δ : Type u_4}
[measurable_space α]
[measurable_singleton_class α]
{a : α}
(f : α → δ) :
f =ᵐ[measure_theory.measure.dirac a] function.const α (f a)
@[class]
structure
measure_theory.is_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
Prop
A measure μ is called finite if μ univ < ∞.
Instances of this typeclass
- measure_theory.is_finite_measure_of_is_empty
- measure_theory.is_probability_measure.to_is_finite_measure
- measure_theory.restrict.is_finite_measure
- measure_theory.is_finite_measure_restrict
- measure_theory.is_finite_measure_zero
- measure_theory.is_finite_measure_add
- measure_theory.is_finite_measure_smul_nnreal
- measure_theory.is_finite_measure_smul_of_nnreal_tower
- measure_theory.measure.is_finite_measure_map
- measure_theory.measure.count.is_finite_measure
- measure_theory.is_finite_measure_trim
- measure_theory.measure.prod.measure_theory.is_finite_measure
- measure_theory.measure.fst.measure_theory.is_finite_measure
- measure_theory.measure.snd.measure_theory.is_finite_measure
- real.is_finite_measure_restrict_Icc
- real.is_finite_measure_restrict_Ico
- real.is_finite_measure_restrict_Ioc
- real.is_finite_measure_restrict_Ioo
- measure_theory.signed_measure.to_measure_of_zero_le_finite
- measure_theory.signed_measure.to_measure_of_le_zero_finite
- measure_theory.measure.is_finite_measure_sub
- measure_theory.jordan_decomposition.pos_part_finite
- measure_theory.jordan_decomposition.neg_part_finite
- measure_theory.measure.singular_part.measure_theory.is_finite_measure
- measure_theory.measure.with_density.measure_theory.is_finite_measure
- measure_theory.measure.restrict.measure_theory.is_finite_measure
- probability_theory.is_finite_kernel.is_finite_measure
- add_circle.is_finite_measure
- unit_add_circle.is_finite_measure
- measure_theory.finite_measure.is_finite_measure
theorem
measure_theory.not_is_finite_measure_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
@[protected, instance]
def
measure_theory.restrict.is_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
{s : set α}
(μ : measure_theory.measure α)
[hs : fact (⇑μ s < ⊤)] :
theorem
measure_theory.measure_lt_top
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(s : set α) :
@[protected, instance]
def
measure_theory.is_finite_measure_restrict
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(s : set α)
[h : measure_theory.is_finite_measure μ] :
theorem
measure_theory.measure_ne_top
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(s : set α) :
theorem
measure_theory.measure_compl_le_add_of_le_add
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
[measure_theory.is_finite_measure μ]
(hs : measurable_set s)
(ht : measurable_set t)
{ε : ennreal}
(h : ⇑μ s ≤ ⇑μ t + ε) :
theorem
measure_theory.measure_compl_le_add_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s t : set α}
[measure_theory.is_finite_measure μ]
(hs : measurable_set s)
(ht : measurable_set t)
{ε : ennreal} :
def
measure_theory.measure_univ_nnreal
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
The measure of the whole space with respect to a finite measure, considered as ℝ≥0.
Equations
@[simp]
theorem
measure_theory.coe_measure_univ_nnreal
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ] :
@[protected, instance]
@[protected, instance]
def
measure_theory.is_finite_measure_of_is_empty
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[is_empty α] :
@[simp]
@[protected, instance]
def
measure_theory.is_finite_measure_add
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
[measure_theory.is_finite_measure ν] :
@[protected, instance]
def
measure_theory.is_finite_measure_smul_nnreal
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{r : nnreal} :
@[protected, instance]
def
measure_theory.is_finite_measure_smul_of_nnreal_tower
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{R : Type u_2}
[has_smul R nnreal]
[has_smul R ennreal]
[is_scalar_tower R nnreal ennreal]
[is_scalar_tower R ennreal ennreal]
[measure_theory.is_finite_measure μ]
{r : R} :
theorem
measure_theory.is_finite_measure_of_le
{α : Type u_1}
{m0 : measurable_space α}
{ν : measure_theory.measure α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(h : ν ≤ μ) :
@[instance]
theorem
measure_theory.measure.is_finite_measure_map
{α : Type u_1}
{β : Type u_2}
[measurable_space β]
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(f : α → β) :
@[simp]
theorem
measure_theory.measure_univ_nnreal_eq_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ] :
measure_theory.measure_univ_nnreal μ = 0 ↔ μ = 0
theorem
measure_theory.measure_univ_nnreal_pos
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
(hμ : μ ≠ 0) :
theorem
measure_theory.measure.le_of_add_le_add_left
{α : Type u_1}
{m0 : measurable_space α}
{μ ν₁ ν₂ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
(A2 : μ + ν₁ ≤ μ + ν₂) :
ν₁ ≤ ν₂
le_of_add_le_add_left is normally applicable to ordered_cancel_add_comm_monoid,
but it holds for measures with the additional assumption that μ is finite.
theorem
measure_theory.summable_measure_to_real
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[hμ : measure_theory.is_finite_measure μ]
{f : ℕ → set α}
(hf₁ : ∀ (i : ℕ), measurable_set (f i))
(hf₂ : pairwise (disjoint on f)) :
theorem
measure_theory.ae_eq_univ_iff_measure_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_finite_measure μ]
(hs : measure_theory.null_measurable_set s μ) :
theorem
measure_theory.ae_iff_measure_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{p : α → Prop}
(hp : measure_theory.null_measurable_set {a : α | p a} μ) :
theorem
measure_theory.ae_mem_iff_measure_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{s : set α}
(hs : measure_theory.null_measurable_set s μ) :
@[protected, instance]
@[class]
structure
measure_theory.is_probability_measure
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
Prop
A measure μ is called a probability measure if μ univ = 1.
Instances of this typeclass
- measure_theory.measure.dirac.is_probability_measure
- measure_theory.measure.prod.measure_theory.is_probability_measure
- measure_theory.measure.fst.measure_theory.is_probability_measure
- measure_theory.measure.snd.measure_theory.is_probability_measure
- pmf.to_measure.is_probability_measure
- probability_theory.is_markov_kernel.is_probability_measure'
- probability_theory.measure.measure_theory.is_probability_measure
- measure_theory.probability_measure.measure_theory.is_probability_measure
- add_circle.haar_add_circle.measure_theory.is_probability_measure
- theorems_100.measure_theory.measure_space.volume.measure_theory.is_probability_measure
@[protected, instance]
def
measure_theory.is_probability_measure.to_is_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_probability_measure μ] :
theorem
measure_theory.is_probability_measure.ne_zero
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_probability_measure μ] :
μ ≠ 0
@[protected, instance]
def
measure_theory.is_probability_measure.ae_ne_bot
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_probability_measure μ] :
@[protected, instance]
def
measure_theory.measure.dirac.is_probability_measure
{α : Type u_1}
[measurable_space α]
{x : α} :
theorem
measure_theory.prob_add_prob_compl
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ]
(h : measurable_set s) :
theorem
measure_theory.prob_le_one
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ] :
theorem
measure_theory.is_probability_measure_smul
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
(h : μ ≠ 0) :
theorem
measure_theory.is_probability_measure_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
[measure_theory.is_probability_measure μ]
{f : α → β}
(hf : ae_measurable f μ) :
@[simp]
theorem
measure_theory.one_le_prob_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ] :
theorem
measure_theory.prob_compl_eq_one_sub
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ]
(hs : measurable_set s) :
Note that this is not quite as useful as it looks because the measure takes values in ℝ≥0∞.
Thus the subtraction appearing is the truncated subtraction of ℝ≥0∞, rather than the
better-behaved subtraction of ℝ.
@[simp]
theorem
measure_theory.prob_compl_eq_zero_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ]
(hs : measurable_set s) :
@[simp]
theorem
measure_theory.prob_compl_eq_one_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.is_probability_measure μ]
(hs : measurable_set s) :
@[class]
structure
measure_theory.has_no_atoms
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
Prop
Measure μ has no atoms if the measure of each singleton is zero.
NB: Wikipedia assumes that for any measurable set s with positive μ-measure,
there exists a measurable t ⊆ s such that 0 < μ t < μ s. While this implies μ {x} = 0,
the converse is not true.
theorem
set.subsingleton.measure_zero
{α : Type u_1}
{m : measurable_space α}
{s : set α}
(hs : s.subsingleton)
(μ : measure_theory.measure α)
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.measure.restrict_singleton'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
{a : α} :
@[protected, instance]
def
measure_theory.measure.restrict.has_no_atoms
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
(s : set α) :
theorem
set.countable.measure_zero
{α : Type u_1}
{m : measurable_space α}
{s : set α}
(h : s.countable)
(μ : measure_theory.measure α)
[measure_theory.has_no_atoms μ] :
theorem
set.countable.ae_not_mem
{α : Type u_1}
{m : measurable_space α}
{s : set α}
(h : s.countable)
(μ : measure_theory.measure α)
[measure_theory.has_no_atoms μ] :
theorem
set.finite.measure_zero
{α : Type u_1}
{m : measurable_space α}
{s : set α}
(h : s.finite)
(μ : measure_theory.measure α)
[measure_theory.has_no_atoms μ] :
theorem
finset.measure_zero
{α : Type u_1}
{m : measurable_space α}
(s : finset α)
(μ : measure_theory.measure α)
[measure_theory.has_no_atoms μ] :
theorem
measure_theory.insert_ae_eq_self
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
(a : α)
(s : set α) :
has_insert.insert a s =ᵐ[μ] s
theorem
measure_theory.Iio_ae_eq_Iic
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a : α} :
theorem
measure_theory.Ioi_ae_eq_Ici
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a : α} :
theorem
measure_theory.Ioo_ae_eq_Ioc
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.Ioc_ae_eq_Icc
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.Ioo_ae_eq_Ico
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.Ioo_ae_eq_Icc
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.Ico_ae_eq_Icc
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.Ico_ae_eq_Ioc
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[partial_order α]
{a b : α} :
theorem
measure_theory.uIoc_ae_eq_interval
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.has_no_atoms μ]
[linear_order α]
{a b : α} :
theorem
measure_theory.ite_ae_eq_of_measure_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{γ : Type u_2}
(f g : α → γ)
(s : set α)
(hs_zero : ⇑μ s = 0) :
theorem
measure_theory.ite_ae_eq_of_measure_compl_zero
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{γ : Type u_2}
(f g : α → γ)
(s : set α)
(hs_zero : ⇑μ sᶜ = 0) :
def
measure_theory.measure.finite_at_filter
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(f : filter α) :
Prop
A measure is called finite at filter f if it is finite at some set s ∈ f.
Equivalently, it is eventually finite at s in f.small_sets.
theorem
measure_theory.measure.finite_at_filter_of_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
(f : filter α) :
μ.finite_at_filter f
theorem
measure_theory.measure.finite_at_filter.exists_mem_basis
{α : Type u_1}
{ι : Type u_5}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : filter α}
(hμ : μ.finite_at_filter f)
{p : ι → Prop}
{s : ι → set α}
(hf : f.has_basis p s) :
theorem
measure_theory.measure.finite_at_bot
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
@[nolint]
structure
measure_theory.measure.finite_spanning_sets_in
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(C : set (set α)) :
Type u_1
- set : ℕ → set α
- set_mem : ∀ (i : ℕ), self.set i ∈ C
- finite : ∀ (i : ℕ), ⇑μ (self.set i) < ⊤
- spanning : (⋃ (i : ℕ), self.set i) = set.univ
μ has finite spanning sets in C if there is a countable sequence of sets in C that have
finite measures. This structure is a type, which is useful if we want to record extra properties
about the sets, such as that they are monotone.
sigma_finite is defined in terms of this: μ is σ-finite if there exists a sequence of
finite spanning sets in the collection of all measurable sets.
Instances for measure_theory.measure.finite_spanning_sets_in
- measure_theory.measure.finite_spanning_sets_in.has_sizeof_inst
@[class]
structure
measure_theory.sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α) :
Prop
- out' : nonempty (μ.finite_spanning_sets_in set.univ)
A measure μ is called σ-finite if there is a countable collection of sets
{ A i | i ∈ ℕ } such that μ (A i) < ∞ and ⋃ i, A i = s.
Instances of this typeclass
- measure_theory.is_finite_measure.to_sigma_finite
- measure_theory.sigma_finite_of_locally_finite
- measure_theory.measure.regular.sigma_finite
- measure_theory.measure.is_haar_measure.sigma_finite
- measure_theory.measure.is_add_haar_measure.sigma_finite
- measure_theory.restrict.sigma_finite
- measure_theory.sum.sigma_finite
- measure_theory.add.sigma_finite
- measure_theory.measure.prod.sigma_finite
- measure_theory.measure.inv.measure_theory.sigma_finite
- measure_theory.measure.neg.measure_theory.sigma_finite
- measure_theory.measure.sigma_finite_tprod
- measure_theory.measure.pi.sigma_finite
- measure_theory.measure.sigma_finite_haar_measure
- measure_theory.measure.sigma_finite_add_haar_measure
- measure_theory.ae_fin_strongly_measurable.sigma_finite_restrict
- measure_theory.measure.singular_part.measure_theory.sigma_finite
- measure_theory.measure.with_density.measure_theory.sigma_finite
- measure_theory.sigma_finite_of_sigma_finite_filtration
- measure_theory.is_stopping_time.sigma_finite_stopping_time
- measure_theory.is_stopping_time.sigma_finite_stopping_time_of_le
theorem
measure_theory.sigma_finite_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.sigma_finite.out
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(h : measure_theory.sigma_finite μ) :
def
measure_theory.measure.to_finite_spanning_sets_in
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[h : measure_theory.sigma_finite μ] :
μ.finite_spanning_sets_in {s : set α | measurable_set s}
If μ is σ-finite it has finite spanning sets in the collection of all measurable sets.
def
measure_theory.spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(i : ℕ) :
set α
A noncomputable way to get a monotone collection of sets that span univ and have finite
measure using classical.some. This definition satisfies monotonicity in addition to all other
properties in sigma_finite.
Equations
theorem
measure_theory.monotone_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ] :
theorem
measure_theory.measurable_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(i : ℕ) :
theorem
measure_theory.measure_spanning_sets_lt_top
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(i : ℕ) :
⇑μ (measure_theory.spanning_sets μ i) < ⊤
theorem
measure_theory.Union_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ] :
theorem
measure_theory.is_countably_spanning_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ] :
noncomputable
def
measure_theory.spanning_sets_index
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(x : α) :
spanning_sets_index μ x is the least n : ℕ such that x ∈ spanning_sets μ n.
Equations
theorem
measure_theory.measurable_spanning_sets_index
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ] :
theorem
measure_theory.preimage_spanning_sets_index_singleton
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(n : ℕ) :
theorem
measure_theory.spanning_sets_index_eq_iff
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
{x : α}
{n : ℕ} :
measure_theory.spanning_sets_index μ x = n ↔ x ∈ disjointed (measure_theory.spanning_sets μ) n
theorem
measure_theory.mem_disjointed_spanning_sets_index
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(x : α) :
theorem
measure_theory.mem_spanning_sets_index
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(x : α) :
theorem
measure_theory.mem_spanning_sets_of_index_le
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(x : α)
{n : ℕ}
(hn : measure_theory.spanning_sets_index μ x ≤ n) :
theorem
measure_theory.eventually_mem_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(x : α) :
∀ᶠ (n : ℕ) in filter.at_top, x ∈ measure_theory.spanning_sets μ n
theorem
measure_theory.measure.supr_restrict_spanning_sets
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.sigma_finite μ]
(hs : measurable_set s) :
theorem
measure_theory.measure.exists_subset_measure_lt_top
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
[measure_theory.sigma_finite μ]
{r : ennreal}
(hs : measurable_set s)
(h's : r < ⇑μ s) :
In a σ-finite space, any measurable set of measure > r contains a measurable subset of
finite measure > r.
theorem
measure_theory.measure.forall_measure_inter_spanning_sets_eq_zero
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
(s : set α) :
A set in a σ-finite space has zero measure if and only if its intersection with all members of the countable family of finite measure spanning sets has zero measure.
theorem
measure_theory.measure.exists_measure_inter_spanning_sets_pos
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
(s : set α) :
A set in a σ-finite space has positive measure if and only if its intersection with some member of the countable family of finite measure spanning sets has positive measure.
theorem
measure_theory.measure.finite_const_le_meas_of_disjoint_Union
{α : Type u_1}
{ι : Type u_2}
[measurable_space α]
(μ : measure_theory.measure α)
{ε : ennreal}
(ε_pos : 0 < ε)
{As : ι → set α}
(As_mble : ∀ (i : ι), measurable_set (As i))
(As_disj : pairwise (disjoint on As))
(Union_As_finite : ⇑μ (⋃ (i : ι), As i) ≠ ⊤) :
If the union of disjoint measurable sets has finite measure, then there are only finitely many members of the union whose measure exceeds any given positive number.
theorem
measure_theory.measure.countable_meas_pos_of_disjoint_of_meas_Union_ne_top
{α : Type u_1}
{ι : Type u_2}
[measurable_space α]
(μ : measure_theory.measure α)
{As : ι → set α}
(As_mble : ∀ (i : ι), measurable_set (As i))
(As_disj : pairwise (disjoint on As))
(Union_As_finite : ⇑μ (⋃ (i : ι), As i) ≠ ⊤) :
If the union of disjoint measurable sets has finite measure, then there are only countably many members of the union whose measure is positive.
theorem
measure_theory.measure.countable_meas_pos_of_disjoint_Union
{α : Type u_1}
{ι : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
{As : ι → set α}
(As_mble : ∀ (i : ι), measurable_set (As i))
(As_disj : pairwise (disjoint on As)) :
In a σ-finite space, among disjoint measurable sets, only countably many can have positive measure.
theorem
measure_theory.measure.countable_meas_level_set_pos
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
[measurable_space β]
[measurable_singleton_class β]
{g : α → β}
(g_mble : measurable g) :
theorem
measure_theory.measure.measure_to_measurable_inter_of_cover
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
{t : set α}
{v : ℕ → set α}
(hv : t ⊆ ⋃ (n : ℕ), v n)
(h'v : ∀ (n : ℕ), ⇑μ (t ∩ v n) ≠ ⊤) :
If a set t is covered by a countable family of finite measure sets, then its measurable
superset to_measurable μ t (which has the same measure as t) satisfies,
for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (t ∩ s).
theorem
measure_theory.measure.measure_to_measurable_inter_of_sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
{s : set α}
(hs : measurable_set s)
(t : set α) :
The measurable superset to_measurable μ t of t (which has the same measure as t)
satisfies, for any measurable set s, the equality μ (to_measurable μ t ∩ s) = μ (t ∩ s).
This only holds when μ is σ-finite. For a version without this assumption (but requiring
that t has finite measure), see measure_to_measurable_inter.
@[simp]
theorem
measure_theory.measure.restrict_to_measurable_of_sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
(s : set α) :
μ.restrict (measure_theory.to_measurable μ s) = μ.restrict s
@[protected]
def
measure_theory.measure.finite_spanning_sets_in.mono'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{C D : set (set α)}
(h : μ.finite_spanning_sets_in C)
(hC : C ∩ {s : set α | ⇑μ s < ⊤} ⊆ D) :
If μ has finite spanning sets in C and C ∩ {s | μ s < ∞} ⊆ D then μ has finite spanning
sets in D.
@[protected]
def
measure_theory.measure.finite_spanning_sets_in.mono
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{C D : set (set α)}
(h : μ.finite_spanning_sets_in C)
(hC : C ⊆ D) :
If μ has finite spanning sets in C and C ⊆ D then μ has finite spanning sets in D.
@[protected]
theorem
measure_theory.measure.finite_spanning_sets_in.sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{C : set (set α)}
(h : μ.finite_spanning_sets_in C) :
If μ has finite spanning sets in the collection of measurable sets C, then μ is σ-finite.
@[protected]
theorem
measure_theory.measure.finite_spanning_sets_in.ext
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{C : set (set α)}
(hA : m0 = measurable_space.generate_from C)
(hC : is_pi_system C)
(h : μ.finite_spanning_sets_in C)
(h_eq : ∀ (s : set α), s ∈ C → ⇑μ s = ⇑ν s) :
μ = ν
An extensionality for measures. It is ext_of_generate_from_of_Union formulated in terms of
finite_spanning_sets_in.
@[protected]
theorem
measure_theory.measure.finite_spanning_sets_in.is_countably_spanning
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{C : set (set α)}
(h : μ.finite_spanning_sets_in C) :
def
measure_theory.measure.finite_spanning_sets_in.of_le
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(h : ν ≤ μ)
{C : set (set α)}
(S : μ.finite_spanning_sets_in C) :
Given measures μ, ν where ν ≤ μ, finite_spanning_sets_in.of_le provides the induced
finite_spanning_set with respect to ν from a finite_spanning_set with respect to μ.
theorem
measure_theory.measure.sigma_finite_of_le
{α : Type u_1}
{m0 : measurable_space α}
{ν : measure_theory.measure α}
(μ : measure_theory.measure α)
[hs : measure_theory.sigma_finite μ]
(h : ν ≤ μ) :
@[protected, instance]
def
measure_theory.is_finite_measure.to_sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ] :
Every finite measure is σ-finite.
@[protected, instance]
def
measure_theory.restrict.sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
(s : set α) :
@[protected, instance]
def
measure_theory.sum.sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
{ι : Type u_2}
[finite ι]
(μ : ι → measure_theory.measure α)
[∀ (i : ι), measure_theory.sigma_finite (μ i)] :
@[protected, instance]
def
measure_theory.add.sigma_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : measure_theory.measure α)
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite ν] :
measure_theory.sigma_finite (μ + ν)
theorem
measure_theory.sigma_finite.of_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
(μ : measure_theory.measure α)
{f : α → β}
(hf : ae_measurable f μ)
(h : measure_theory.sigma_finite (measure_theory.measure.map f μ)) :
theorem
measurable_equiv.sigma_finite_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
[measurable_space β]
{μ : measure_theory.measure α}
(f : α ≃ᵐ β)
(h : measure_theory.sigma_finite μ) :
theorem
measure_theory.ae_of_forall_measure_lt_top_ae_restrict'
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(ν : measure_theory.measure α)
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite ν]
(P : α → Prop)
(h : ∀ (s : set α), measurable_set s → ⇑μ s < ⊤ → ⇑ν s < ⊤ → (∀ᵐ (x : α) ∂μ.restrict s, P x)) :
Similar to ae_of_forall_measure_lt_top_ae_restrict, but where you additionally get the
hypothesis that another σ-finite measure has finite values on s.
theorem
measure_theory.ae_of_forall_measure_lt_top_ae_restrict
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
(P : α → Prop)
(h : ∀ (s : set α), measurable_set s → ⇑μ s < ⊤ → (∀ᵐ (x : α) ∂μ.restrict s, P x)) :
To prove something for almost all x w.r.t. a σ-finite measure, it is sufficient to show that
this holds almost everywhere in sets where the measure has finite value.
@[class]
structure
measure_theory.is_locally_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α) :
Prop
- finite_at_nhds : ∀ (x : α), μ.finite_at_filter (nhds x)
A measure is called locally finite if it is finite in some neighborhood of each point.
Instances of this typeclass
- measure_theory.is_finite_measure.to_is_locally_finite_measure
- measure_theory.measure.is_locally_finite_measure_of_is_haar_measure
- measure_theory.measure.is_locally_finite_measure_of_is_add_haar_measure
- measure_theory.is_locally_finite_measure_smul_nnreal
- measure_theory.measure.pi.measure_theory.is_locally_finite_measure
- stieltjes_function.measure.measure_theory.is_locally_finite_measure
- real.locally_finite_volume
- measure_theory.measure.singular_part.measure_theory.is_locally_finite_measure
- measure_theory.measure.with_density.measure_theory.is_locally_finite_measure
- measure_theory.measure.alternating_map.measure.measure_theory.is_locally_finite_measure
@[protected, instance]
def
measure_theory.is_finite_measure.to_is_locally_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ] :
theorem
measure_theory.measure.finite_at_nhds
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ]
(x : α) :
μ.finite_at_filter (nhds x)
theorem
measure_theory.measure.smul_finite
{α : Type u_1}
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure μ]
{c : ennreal}
(hc : c ≠ ⊤) :
theorem
measure_theory.measure.exists_is_open_measure_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ]
(x : α) :
@[protected, instance]
def
measure_theory.is_locally_finite_measure_smul_nnreal
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ]
(c : nnreal) :
@[protected]
theorem
measure_theory.measure.is_topological_basis_is_open_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ] :
@[class]
structure
measure_theory.is_finite_measure_on_compacts
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α) :
Prop
A measure μ is finite on compacts if any compact set K satisfies μ K < ∞.
Instances of this typeclass
- is_finite_measure_on_compacts_of_is_locally_finite_measure
- measure_theory.measure.regular.to_is_finite_measure_on_compacts
- measure_theory.measure.is_add_haar_measure.to_is_finite_measure_on_compacts
- measure_theory.measure.is_haar_measure.to_is_finite_measure_on_compacts
- measure_theory.measure.prod.measure_theory.is_finite_measure_on_compacts
- measure_theory.measure.pi.is_finite_measure_on_compacts
theorem
is_compact.measure_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure_on_compacts μ]
⦃K : set α⦄
(hK : is_compact K) :
A compact subset has finite measure for a measure which is finite on compacts.
theorem
metric.bounded.measure_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[pseudo_metric_space α]
[proper_space α]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure_on_compacts μ]
⦃s : set α⦄
(hs : metric.bounded s) :
A bounded subset has finite measure for a measure which is finite on compact sets, in a proper space.
theorem
measure_theory.measure_closed_ball_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[pseudo_metric_space α]
[proper_space α]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure_on_compacts μ]
{x : α}
{r : ℝ} :
⇑μ (metric.closed_ball x r) < ⊤
theorem
measure_theory.measure_ball_lt_top
{α : Type u_1}
{m0 : measurable_space α}
[pseudo_metric_space α]
[proper_space α]
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure_on_compacts μ]
{x : α}
{r : ℝ} :
⇑μ (metric.ball x r) < ⊤
@[protected]
theorem
measure_theory.is_finite_measure_on_compacts.smul
{α : Type u_1}
{m0 : measurable_space α}
[topological_space α]
(μ : measure_theory.measure α)
[measure_theory.is_finite_measure_on_compacts μ]
{c : ennreal}
(hc : c ≠ ⊤) :
theorem
measure_theory.compact_space.is_finite_measure
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[compact_space α]
[measure_theory.is_finite_measure_on_compacts μ] :
Note this cannot be an instance because it would form a typeclass loop with
is_finite_measure_on_compacts_of_is_locally_finite_measure.
@[protected, instance]
def
measure_theory.sigma_finite_of_locally_finite
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[topological_space.second_countable_topology α]
[measure_theory.is_locally_finite_measure μ] :
theorem
measure_theory.is_locally_finite_measure_of_is_finite_measure_on_compacts
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[locally_compact_space α]
[measure_theory.is_finite_measure_on_compacts μ] :
A measure which is finite on compact sets in a locally compact space is locally finite. Not registered as an instance to avoid a loop with the other direction.
theorem
measure_theory.exists_pos_preimage_ball
{α : Type u_1}
{δ : Type u_4}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[pseudo_metric_space δ]
(f : α → δ)
(x : δ)
(hμ : μ ≠ 0) :
theorem
measure_theory.exists_pos_ball
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[pseudo_metric_space α]
(x : α)
(hμ : μ ≠ 0) :
theorem
measure_theory.null_of_locally_null
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[topological_space.second_countable_topology α]
(s : set α)
(hs : ∀ (x : α), x ∈ s → (∃ (u : set α) (H : u ∈ nhds_within x s), ⇑μ u = 0)) :
If a set has zero measure in a neighborhood of each of its points, then it has zero measure in a second-countable space.
theorem
measure_theory.exists_mem_forall_mem_nhds_within_pos_measure
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[topological_space.second_countable_topology α]
{s : set α}
(hs : ⇑μ s ≠ 0) :
theorem
measure_theory.exists_ne_forall_mem_nhds_pos_measure_preimage
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{β : Type u_2}
[topological_space β]
[t1_space β]
[topological_space.second_countable_topology β]
[nonempty β]
{f : α → β}
(h : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b) :
theorem
measure_theory.ext_on_measurable_space_of_generate_finite
{α : Type u_1}
(m₀ : measurable_space α)
{μ ν : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
(C : set (set α))
(hμν : ∀ (s : set α), s ∈ C → ⇑μ s = ⇑ν s)
{m : measurable_space α}
(h : m ≤ m₀)
(hA : m = measurable_space.generate_from C)
(hC : is_pi_system C)
(h_univ : ⇑μ set.univ = ⇑ν set.univ)
{s : set α}
(hs : measurable_set s) :
If two finite measures give the same mass to the whole space and coincide on a π-system made of measurable sets, then they coincide on all sets in the σ-algebra generated by the π-system.
theorem
measure_theory.ext_of_generate_finite
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
(C : set (set α))
(hA : m0 = measurable_space.generate_from C)
(hC : is_pi_system C)
[measure_theory.is_finite_measure μ]
(hμν : ∀ (s : set α), s ∈ C → ⇑μ s = ⇑ν s)
(h_univ : ⇑μ set.univ = ⇑ν set.univ) :
μ = ν
Two finite measures are equal if they are equal on the π-system generating the σ-algebra
(and univ).
@[protected]
def
measure_theory.measure.finite_spanning_sets_in.disjointed
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(S : μ.finite_spanning_sets_in {s : set α | measurable_set s}) :
μ.finite_spanning_sets_in {s : set α | measurable_set s}
Given S : μ.finite_spanning_sets_in {s | measurable_set s},
finite_spanning_sets_in.disjointed provides a finite_spanning_sets_in {s | measurable_set s}
such that its underlying sets are pairwise disjoint.
Equations
- S.disjointed = {set := disjointed S.set, set_mem := _, finite := _, spanning := _}
theorem
measure_theory.measure.finite_spanning_sets_in.disjointed_set_eq
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(S : μ.finite_spanning_sets_in {s : set α | measurable_set s}) :
S.disjointed.set = disjointed S.set
theorem
measure_theory.measure.exists_eq_disjoint_finite_spanning_sets_in
{α : Type u_1}
{m0 : measurable_space α}
(μ ν : measure_theory.measure α)
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite ν] :
∃ (S : μ.finite_spanning_sets_in {s : set α | measurable_set s}) (T : ν.finite_spanning_sets_in {s : set α | measurable_set s}), S.set = T.set ∧ pairwise (disjoint on S.set)
theorem
measure_theory.measure.finite_at_filter.filter_mono
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : filter α}
(h : f ≤ g) :
μ.finite_at_filter g → μ.finite_at_filter f
theorem
measure_theory.measure.finite_at_filter.inf_of_left
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : filter α}
(h : μ.finite_at_filter f) :
μ.finite_at_filter (f ⊓ g)
theorem
measure_theory.measure.finite_at_filter.inf_of_right
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : filter α}
(h : μ.finite_at_filter g) :
μ.finite_at_filter (f ⊓ g)
@[simp]
theorem
measure_theory.measure.finite_at_filter.inf_ae_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : filter α} :
μ.finite_at_filter (f ⊓ μ.ae) ↔ μ.finite_at_filter f
theorem
measure_theory.measure.finite_at_filter.of_inf_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : filter α} :
μ.finite_at_filter (f ⊓ μ.ae) → μ.finite_at_filter f
Alias of the forward direction of measure_theory.measure.finite_at_filter.inf_ae_iff.
theorem
measure_theory.measure.finite_at_filter.filter_mono_ae
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : filter α}
(h : f ⊓ μ.ae ≤ g)
(hg : μ.finite_at_filter g) :
μ.finite_at_filter f
@[protected]
theorem
measure_theory.measure.finite_at_filter.measure_mono
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{f : filter α}
(h : μ ≤ ν) :
ν.finite_at_filter f → μ.finite_at_filter f
@[protected]
theorem
measure_theory.measure.finite_at_filter.mono
{α : Type u_1}
{m0 : measurable_space α}
{μ ν : measure_theory.measure α}
{f g : filter α}
(hf : f ≤ g)
(hμ : μ ≤ ν) :
ν.finite_at_filter g → μ.finite_at_filter f
@[protected]
theorem
measure_theory.measure.finite_at_filter.eventually
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : filter α}
(h : μ.finite_at_filter f) :
theorem
measure_theory.measure.finite_at_filter.filter_sup
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : filter α} :
μ.finite_at_filter f → μ.finite_at_filter g → μ.finite_at_filter (f ⊔ g)
theorem
measure_theory.measure.finite_at_nhds_within
{α : Type u_1}
[topological_space α]
{m0 : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ]
(x : α)
(s : set α) :
μ.finite_at_filter (nhds_within x s)
@[simp]
theorem
measure_theory.measure.finite_at_principal
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α} :
μ.finite_at_filter (filter.principal s) ↔ ⇑μ s < ⊤
theorem
measure_theory.measure.is_locally_finite_measure_of_le
{α : Type u_1}
[topological_space α]
{m : measurable_space α}
{μ ν : measure_theory.measure α}
[H : measure_theory.is_locally_finite_measure μ]
(h : ν ≤ μ) :
theorem
measurable_embedding.map_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(μ : measure_theory.measure α)
(s : set β) :
⇑(measure_theory.measure.map f μ) s = ⇑μ (f ⁻¹' s)
theorem
measurable_embedding.map_comap
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(μ : measure_theory.measure β) :
measure_theory.measure.map f (measure_theory.measure.comap f μ) = μ.restrict (set.range f)
theorem
measurable_embedding.comap_apply
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(μ : measure_theory.measure β)
(s : set α) :
⇑(measure_theory.measure.comap f μ) s = ⇑μ (f '' s)
theorem
measurable_embedding.ae_map_iff
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
{p : β → Prop}
{μ : measure_theory.measure α} :
theorem
measurable_embedding.restrict_map
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(μ : measure_theory.measure α)
(s : set β) :
(measure_theory.measure.map f μ).restrict s = measure_theory.measure.map f (μ.restrict (f ⁻¹' s))
@[protected]
theorem
measurable_embedding.comap_preimage
{α : Type u_1}
{β : Type u_2}
{m0 : measurable_space α}
{m1 : measurable_space β}
{f : α → β}
(hf : measurable_embedding f)
(μ : measure_theory.measure β)
{s : set β}
(hs : measurable_set s) :
theorem
comap_subtype_coe_apply
{α : Type u_1}
{m0 : measurable_space α}
{s : set α}
(hs : measurable_set s)
(μ : measure_theory.measure α)
(t : set ↥s) :
theorem
map_comap_subtype_coe
{α : Type u_1}
{m0 : measurable_space α}
{s : set α}
(hs : measurable_set s)
(μ : measure_theory.measure α) :
theorem
ae_restrict_iff_subtype
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
{p : α → Prop} :
@[protected, instance]
noncomputable
def
set_coe.measure_space
{α : Type u_1}
[measure_theory.measure_space α]
(s : set α) :
theorem
measurable_set.map_coe_volume
{α : Type u_1}
[measure_theory.measure_space α]
{s : set α}
(hs : measurable_set s) :
theorem
volume_image_subtype_coe
{α : Type u_1}
[measure_theory.measure_space α]
{s : set α}
(hs : measurable_set s)
(t : set ↥s) :
@[simp]
theorem
volume_preimage_coe
{α : Type u_1}
[measure_theory.measure_space α]
{s t : set α}
(hs : measure_theory.null_measurable_set s measure_theory.measure_space.volume)
(ht : measurable_set t) :
Interactions of measurable equivalences and measures
@[protected]
theorem
measurable_equiv.map_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
(f : α ≃ᵐ β)
(s : set β) :
If we map a measure along a measurable equivalence, we can compute the measure on all sets (not just the measurable ones).
@[simp]
theorem
measurable_equiv.map_symm_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
(e : α ≃ᵐ β) :
measure_theory.measure.map ⇑(e.symm) (measure_theory.measure.map ⇑e μ) = μ
@[simp]
theorem
measurable_equiv.map_map_symm
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{ν : measure_theory.measure β}
(e : α ≃ᵐ β) :
measure_theory.measure.map ⇑e (measure_theory.measure.map ⇑(e.symm) ν) = ν
theorem
measurable_equiv.map_measurable_equiv_injective
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
(e : α ≃ᵐ β) :
theorem
measurable_equiv.map_apply_eq_iff_map_symm_apply_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
(e : α ≃ᵐ β) :
measure_theory.measure.map ⇑e μ = ν ↔ measure_theory.measure.map ⇑(e.symm) ν = μ
theorem
measurable_equiv.restrict_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
(e : α ≃ᵐ β)
(s : set β) :
(measure_theory.measure.map ⇑e μ).restrict s = measure_theory.measure.map ⇑e (μ.restrict (⇑e ⁻¹' s))
theorem
measurable_equiv.map_ae
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
(f : α ≃ᵐ β)
(μ : measure_theory.measure α) :
filter.map ⇑f μ.ae = (measure_theory.measure.map ⇑f μ).ae
theorem
measurable_equiv.quasi_measure_preserving_symm
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
(μ : measure_theory.measure α)
(e : α ≃ᵐ β) :
theorem
measure_theory.outer_measure.to_measure_zero
{α : Type u_1}
[measurable_space α] :
0.to_measure _ = 0
noncomputable
def
measure_theory.measure.trim
{α : Type u_1}
{m m0 : measurable_space α}
(μ : measure_theory.measure α)
(hm : m ≤ m0) :
Restriction of a measure to a sub-sigma algebra.
It is common to see a measure μ on a measurable space structure m0 as being also a measure on
any m ≤ m0. Since measures in mathlib have to be trimmed to the measurable space, μ itself
cannot be a measure on m, hence the definition of μ.trim hm.
This notion is related to outer_measure.trim, see the lemma
to_outer_measure_trim_eq_trim_to_outer_measure.
Equations
- μ.trim hm = μ.to_outer_measure.to_measure _
@[simp]
theorem
measure_theory.trim_eq_self
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α} :
theorem
measure_theory.to_outer_measure_trim_eq_trim_to_outer_measure
{α : Type u_1}
{m m0 : measurable_space α}
(μ : measure_theory.measure α)
(hm : m ≤ m0) :
(μ.trim hm).to_outer_measure = μ.to_outer_measure.trim
@[simp]
theorem
measure_theory.trim_measurable_set_eq
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hm : m ≤ m0)
(hs : measurable_set s) :
theorem
measure_theory.le_trim
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hm : m ≤ m0) :
theorem
measure_theory.measure_eq_zero_of_trim_eq_zero
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hm : m ≤ m0)
(h : ⇑(μ.trim hm) s = 0) :
theorem
measure_theory.measure_trim_to_measurable_eq_zero
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{hm : m ≤ m0}
(hs : ⇑(μ.trim hm) s = 0) :
⇑μ (measure_theory.to_measurable (μ.trim hm) s) = 0
theorem
measure_theory.ae_of_ae_trim
{α : Type u_1}
{m m0 : measurable_space α}
(hm : m ≤ m0)
{μ : measure_theory.measure α}
{P : α → Prop}
(h : ∀ᵐ (x : α) ∂μ.trim hm, P x) :
theorem
measure_theory.ae_eq_of_ae_eq_trim
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_2}
{hm : m ≤ m0}
{f₁ f₂ : α → E}
(h12 : f₁ =ᵐ[μ.trim hm] f₂) :
theorem
measure_theory.ae_le_of_ae_le_trim
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_2}
[has_le E]
{hm : m ≤ m0}
{f₁ f₂ : α → E}
(h12 : f₁ ≤ᵐ[μ.trim hm] f₂) :
theorem
measure_theory.trim_trim
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α}
{m₁ m₂ : measurable_space α}
{hm₁₂ : m₁ ≤ m₂}
{hm₂ : m₂ ≤ m0} :
theorem
measure_theory.restrict_trim
{α : Type u_1}
{m m0 : measurable_space α}
{s : set α}
(hm : m ≤ m0)
(μ : measure_theory.measure α)
(hs : measurable_set s) :
@[protected, instance]
def
measure_theory.is_finite_measure_trim
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.is_finite_measure μ] :
theorem
measure_theory.sigma_finite_trim_mono
{α : Type u_1}
{m m₂ m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
(hm₂ : m₂ ≤ m)
[measure_theory.sigma_finite (μ.trim _)] :
measure_theory.sigma_finite (μ.trim hm)
theorem
measure_theory.sigma_finite_trim_bot_iff
{α : Type u_1}
{m0 : measurable_space α}
{μ : measure_theory.measure α} :
theorem
is_compact.exists_open_superset_measure_lt_top'
{α : Type u_1}
[topological_space α]
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
(h : is_compact s)
(hμ : ∀ (x : α), x ∈ s → μ.finite_at_filter (nhds x)) :
If s is a compact set and μ is finite at 𝓝 x for every x ∈ s, then s admits an open
superset of finite measure.
theorem
is_compact.exists_open_superset_measure_lt_top
{α : Type u_1}
[topological_space α]
[measurable_space α]
{s : set α}
(h : is_compact s)
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ] :
If s is a compact set and μ is a locally finite measure, then s admits an open superset of
finite measure.
theorem
is_compact.measure_lt_top_of_nhds_within
{α : Type u_1}
[topological_space α]
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
(h : is_compact s)
(hμ : ∀ (x : α), x ∈ s → μ.finite_at_filter (nhds_within x s)) :
theorem
is_compact.measure_zero_of_nhds_within
{α : Type u_1}
[topological_space α]
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
(hs : is_compact s) :
@[protected, instance]
def
measure_theory.measure.finite_spanning_sets_in_compact
{α : Type u_1}
[topological_space α]
[sigma_compact_space α]
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ] :
μ.finite_spanning_sets_in {K : set α | is_compact K}
Compact covering of a σ-compact topological space as
measure_theory.measure.finite_spanning_sets_in.
Equations
- μ.finite_spanning_sets_in_compact = {set := compact_covering α _inst_2, set_mem := _, finite := _, spanning := _}
def
measure_theory.measure.finite_spanning_sets_in_open
{α : Type u_1}
[topological_space α]
[sigma_compact_space α]
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ] :
μ.finite_spanning_sets_in {K : set α | is_open K}
A locally finite measure on a σ-compact topological space admits a finite spanning sequence
of open sets.
@[irreducible]
noncomputable
def
measure_theory.measure.finite_spanning_sets_in_open'
{α : Type u_1}
[topological_space α]
[topological_space.second_countable_topology α]
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.is_locally_finite_measure μ] :
μ.finite_spanning_sets_in {K : set α | is_open K}
A locally finite measure on a second countable topological space admits a finite spanning sequence of open sets.
Equations
theorem
measure_Icc_lt_top
{α : Type u_1}
[preorder α]
[topological_space α]
[compact_Icc_space α]
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a b : α} :
theorem
measure_Ico_lt_top
{α : Type u_1}
[preorder α]
[topological_space α]
[compact_Icc_space α]
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a b : α} :
theorem
measure_Ioc_lt_top
{α : Type u_1}
[preorder α]
[topological_space α]
[compact_Icc_space α]
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a b : α} :
theorem
measure_Ioo_lt_top
{α : Type u_1}
[preorder α]
[topological_space α]
[compact_Icc_space α]
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_locally_finite_measure μ]
{a b : α} :
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
piecewise_ae_eq_of_ae_eq_set
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s t : set α}
{f g : α → β}
(hst : s =ᵐ[μ] t) :
theorem
mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
{t : set β}
(ht : 0 ∈ t)
(hs : measurable_set s) :
t ∈ filter.map (s.indicator f) μ.ae ↔ t ∈ filter.map f (μ.restrict s).ae
theorem
map_restrict_ae_le_map_indicator_ae
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : measurable_set s) :
filter.map f (μ.restrict s).ae ≤ filter.map (s.indicator f) μ.ae
theorem
indicator_ae_eq_restrict
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : measurable_set s) :
theorem
indicator_ae_eq_restrict_compl
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : measurable_set s) :
theorem
indicator_ae_eq_of_restrict_compl_ae_eq_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : measurable_set s)
(hf : f =ᵐ[μ.restrict sᶜ] 0) :
theorem
indicator_ae_eq_zero_of_restrict_ae_eq_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : measurable_set s)
(hf : f =ᵐ[μ.restrict s] 0) :
theorem
indicator_ae_eq_of_ae_eq_set
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s t : set α}
{f : α → β}
[has_zero β]
(hst : s =ᵐ[μ] t) :
theorem
indicator_meas_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
(hs : ⇑μ s = 0) :
theorem
ae_eq_restrict_iff_indicator_ae_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{s : set α}
{f : α → β}
[has_zero β]
{g : α → β}
(hs : measurable_set s) :