File: Mathlib/MeasureTheory/Measure/EverywherePos.lean

full_name: MeasureTheory.Measure.IsEverywherePos
code: /-- A set `s` is *everywhere positive* (also called *self-supporting*) with respect to a
measure `μ` if it has positive measure around each of its points, i.e., if all neighborhoods `n`
of points of `s` satisfy `μ (s ∩ n) > 0`. -/
def IsEverywherePos (μ : Measure α) (s : Set α) : Prop 
goal: Prop

full_name: MeasureTheory.Measure.everywherePosSubset
code: /-- * The everywhere positive subset of a set is the subset made of those points all of whose
neighborhoods have positive measure inside the set. -/
def everywherePosSubset (μ : Measure α) (s : Set α) : Set α 
goal: Set α

full_name: MeasureTheory.Measure.everywherePosSubset_subset
code: lemma everywherePosSubset_subset (μ : Measure α) (s : Set α) : μ.everywherePosSubset s ⊆ s 
goal: μ.everywherePosSubset s ⊆ s

full_name: MeasureTheory.Measure.exists_isOpen_everywherePosSubset_eq_diff
code: /-- The everywhere positive subset of a set is obtained by removing an open set. -/
lemma exists_isOpen_everywherePosSubset_eq_diff (μ : Measure α) (s : Set α) :
    ∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u 
goal: ∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u

full_name: MeasurableSet.everywherePosSubset
code: protected lemma _root_.MeasurableSet.everywherePosSubset [OpensMeasurableSpace α]
    (hs : MeasurableSet s) :
    MeasurableSet (μ.everywherePosSubset s) 
goal: MeasurableSet (μ.everywherePosSubset s)

full_name: IsClosed.everywherePosSubset
code: protected lemma _root_.IsClosed.everywherePosSubset (hs : IsClosed s) :
    IsClosed (μ.everywherePosSubset s) 
goal: IsClosed (μ.everywherePosSubset s)

full_name: IsCompact.everywherePosSubset
code: protected lemma _root_.IsCompact.everywherePosSubset (hs : IsCompact s) :
    IsCompact (μ.everywherePosSubset s) 
goal: IsCompact (μ.everywherePosSubset s)

full_name: MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset
code: /-- Any compact set contained in `s \ μ.everywherePosSubset s` has zero measure. -/
lemma measure_eq_zero_of_subset_diff_everywherePosSubset
    (hk : IsCompact k) (h'k : k ⊆ s \ μ.everywherePosSubset s) : μ k = 0 
goal: μ k = 0

full_name: MeasureTheory.Measure.everywherePosSubset_ae_eq
code: /-- In a space with an inner regular measure, any measurable set coincides almost everywhere with
its everywhere positive subset. -/
lemma everywherePosSubset_ae_eq [OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
    μ.everywherePosSubset s =ᵐ[μ] s 
goal: μ.everywherePosSubset s =ᵐ[μ] s

full_name: MeasureTheory.Measure.everywherePosSubset_ae_eq_of_measure_ne_top
code: /-- In a space with an inner regular measure for finite measure sets, any measurable set of finite
measure coincides almost everywhere with its everywhere positive subset. -/
lemma everywherePosSubset_ae_eq_of_measure_ne_top
    [OpensMeasurableSpace α] [InnerRegularCompactLTTop μ] (hs : MeasurableSet s) (h's : μ s ≠ ∞) :
    μ.everywherePosSubset s =ᵐ[μ] s 
goal: μ.everywherePosSubset s =ᵐ[μ] s

full_name: MeasureTheory.Measure.isEverywherePos_everywherePosSubset
code: /-- In a space with an inner regular measure, the everywhere positive subset of a measurable set
is itself everywhere positive. This is not obvious as `μ.everywherePosSubset s` is defined as
the points whose neighborhoods intersect `s` along positive measure subsets, but this does not
say they also intersect `μ.everywherePosSubset s` along positive measure subsets. -/
lemma isEverywherePos_everywherePosSubset
    [OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
    μ.IsEverywherePos (μ.everywherePosSubset s) 
goal: μ.IsEverywherePos (μ.everywherePosSubset s)

full_name: MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top
code: /-- In a space with an inner regular measure for finite measure sets, the everywhere positive subset
of a measurable set of finite measure is itself everywhere positive. This is not obvious as
`μ.everywherePosSubset s` is defined as the points whose neighborhoods intersect `s` along positive
measure subsets, but this does not say they also intersect `μ.everywherePosSubset s` along positive
measure subsets. -/
lemma isEverywherePos_everywherePosSubset_of_measure_ne_top
    [OpensMeasurableSpace α] [InnerRegularCompactLTTop μ] (hs : MeasurableSet s) (h's : μ s ≠ ∞) :
    μ.IsEverywherePos (μ.everywherePosSubset s) 
goal: μ.IsEverywherePos (μ.everywherePosSubset s)

full_name: MeasureTheory.Measure.IsEverywherePos.smul_measure
code: lemma IsEverywherePos.smul_measure (hs : IsEverywherePos μ s) {c : ℝ≥0∞} (hc : c ≠ 0) :
    IsEverywherePos (c • μ) s 
goal: (c • μ).IsEverywherePos s

full_name: MeasureTheory.Measure.IsEverywherePos.smul_measure_nnreal
code: lemma IsEverywherePos.smul_measure_nnreal (hs : IsEverywherePos μ s) {c : ℝ≥0} (hc : c ≠ 0) :
    IsEverywherePos (c • μ) s 
goal: (c • μ).IsEverywherePos s

full_name: MeasureTheory.Measure.IsEverywherePos.of_forall_exists_nhds_eq
code: /-- If two measures coincide locally, then a set which is everywhere positive for the former is
also everywhere positive for the latter. -/
lemma IsEverywherePos.of_forall_exists_nhds_eq (hs : IsEverywherePos μ s)
    (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u) : IsEverywherePos ν s 
goal: ν.IsEverywherePos s

full_name: MeasureTheory.Measure.isEverywherePos_iff_of_forall_exists_nhds_eq
code: /-- If two measures coincide locally, then a set is everywhere positive for the former iff it is
everywhere positive for the latter. -/
lemma isEverywherePos_iff_of_forall_exists_nhds_eq (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u) :
    IsEverywherePos ν s ↔ IsEverywherePos μ s 
goal: ν.IsEverywherePos s ↔ μ.IsEverywherePos s

full_name: IsOpen.isEverywherePos
code: /-- An open set is everywhere positive for a measure which is positive on open sets. -/
lemma _root_.IsOpen.isEverywherePos [IsOpenPosMeasure μ] (hs : IsOpen s) : IsEverywherePos μ s 
goal: μ.IsEverywherePos s

full_name: MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant
code: /-- If a compact closed set is everywhere positive with respect to a left-invariant measure on a
topological group, then it is a Gδ set. This is nontrivial, as there is no second-countability or
metrizability assumption in the statement, so a general compact closed set has no reason to be
a countable intersection of open sets. -/
@[to_additive]
lemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant
    {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k) :
    IsGδ k 
goal: IsGδ k

full_name: MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isAddLeftInvariant
code: /-- If a compact closed set is everywhere positive with respect to a left-invariant measure on a
topological group, then it is a Gδ set. This is nontrivial, as there is no second-countability or
metrizability assumption in the statement, so a general compact closed set has no reason to be
a countable intersection of open sets. -/
@[to_additive]
lemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant
    {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k) :
    IsGδ k 
goal: IsGδ k

full_name: MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group
code: /-- **Halmos' theorem: Haar measure is completion regular.** More precisely, any finite measure
set can be approximated from inside by a level set of a continuous function with compact support. -/
@[to_additive innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup]
theorem innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group :
    InnerRegularWRT μ (fun s ↦ ∃ (f : G → ℝ), Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1})
    (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
goal: μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => MeasurableSet s ∧ μ s ≠ ⊤

full_name: MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup
code: /-- **Halmos' theorem: Haar measure is completion regular.** More precisely, any finite measure
set can be approximated from inside by a level set of a continuous function with compact support. -/
@[to_additive innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup]
theorem innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group :
    InnerRegularWRT μ (fun s ↦ ∃ (f : G → ℝ), Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1})
    (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
goal: μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => MeasurableSet s ∧ μ s ≠ ⊤