File: Mathlib/MeasureTheory/Measure/DiracProba.lean

full_name: CompletelyRegularSpace.exists_BCNN
code: lemma CompletelyRegularSpace.exists_BCNN {X : Type*} [TopologicalSpace X] [CompletelyRegularSpace X]
    {K : Set X} (K_closed : IsClosed K) {x : X} (x_notin_K : x ∉ K) :
    ∃ (f : X →ᵇ ℝ≥0), f x = 1 ∧ (∀ y ∈ K, f y = 0) 
goal: ∃ f, f x = 1 ∧ ∀ y ∈ K, f y = 0

full_name: MeasureTheory.diracProba
code: /-- The Dirac delta mass at a point `x : X` as a `ProbabilityMeasure`. -/
noncomputable def diracProba (x : X) : ProbabilityMeasure X 
goal: MeasureTheory.ProbabilityMeasure X

full_name: MeasureTheory.injective_diracProba
code: /-- The assignment `x ↦ diracProba x` is injective if all singletons are measurable. -/
lemma injective_diracProba {X : Type*} [MeasurableSpace X] [MeasurableSpace.SeparatesPoints X] :
    Function.Injective (fun (x : X) ↦ diracProba x) 
goal: Function.Injective fun x => MeasureTheory.diracProba x

full_name: MeasureTheory.diracProba_toMeasure_apply'
code: @[simp] lemma diracProba_toMeasure_apply' (x : X) {A : Set X} (A_mble : MeasurableSet A) :
    (diracProba x).toMeasure A = A.indicator 1 x 
goal: ↑(MeasureTheory.diracProba x) A = A.indicator 1 x

full_name: MeasureTheory.diracProba_toMeasure_apply_of_mem
code: @[simp] lemma diracProba_toMeasure_apply_of_mem {x : X} {A : Set X} (x_in_A : x ∈ A) :
    (diracProba x).toMeasure A = 1 
goal: ↑(MeasureTheory.diracProba x) A = 1

full_name: MeasureTheory.diracProba_toMeasure_apply
code: @[simp] lemma diracProba_toMeasure_apply [MeasurableSingletonClass X] (x : X) (A : Set X) :
    (diracProba x).toMeasure A = A.indicator 1 x 
goal: ↑(MeasureTheory.diracProba x) A = A.indicator 1 x

full_name: MeasureTheory.continuous_diracProba
code: /-- The assignment `x ↦ diracProba x` is continuous `X → ProbabilityMeasure X`. -/
lemma continuous_diracProba : Continuous (fun (x : X) ↦ diracProba x) 
goal: Continuous fun x => MeasureTheory.diracProba x

full_name: MeasureTheory.injective_diracProba_of_T0
code: /-- In a T0 topological space equipped with a sigma algebra which contains all open sets,
the assignment `x ↦ diracProba x` is injective. -/
lemma injective_diracProba_of_T0 [T0Space X] :
    Function.Injective (fun (x : X) ↦ diracProba x) 
goal: Function.Injective fun x => MeasureTheory.diracProba x

full_name: MeasureTheory.not_tendsto_diracProba_of_not_tendsto
code: lemma not_tendsto_diracProba_of_not_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X)
    (h : ¬ Tendsto id L (𝓝 x)) :
    ¬ Tendsto diracProba L (𝓝 (diracProba x)) 
goal: ¬Filter.Tendsto MeasureTheory.diracProba L (𝓝 (MeasureTheory.diracProba x))

full_name: MeasureTheory.tendsto_diracProba_iff_tendsto
code: lemma tendsto_diracProba_iff_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X) :
    Tendsto diracProba L (𝓝 (diracProba x)) ↔ Tendsto id L (𝓝 x) 
goal: Filter.Tendsto MeasureTheory.diracProba L (𝓝 (MeasureTheory.diracProba x)) ↔ Filter.Tendsto id L (𝓝 x)

full_name: MeasureTheory.diracProbaInverse
code: /-- An inverse function to `diracProba` (only really an inverse under hypotheses that
guarantee injectivity of `diracProba`). -/
noncomputable def diracProbaInverse : range (diracProba (X := X)) → X 
goal: ↑(Set.range MeasureTheory.diracProba) → X

full_name: MeasureTheory.diracProba_diracProbaInverse
code: @[simp] lemma diracProba_diracProbaInverse (μ : range (diracProba (X := X))) :
    diracProba (diracProbaInverse μ) = μ 
goal: MeasureTheory.diracProba (MeasureTheory.diracProbaInverse μ) = ↑μ

full_name: MeasureTheory.diracProbaInverse_eq
code: lemma diracProbaInverse_eq [T0Space X] {x : X} {μ : range (diracProba (X := X))}
    (h : μ = diracProba x) :
    diracProbaInverse μ = x 
goal: MeasureTheory.diracProbaInverse μ = x

full_name: MeasureTheory.diracProbaEquiv
code: /-- In a T0 topological space `X`, the assignment `x ↦ diracProba x` is a bijection to its
range in `ProbabilityMeasure X`. -/
noncomputable def diracProbaEquiv [T0Space X] : X ≃ range (diracProba (X := X)) 
goal: X ≃ ↑(Set.range MeasureTheory.diracProba)

full_name: MeasureTheory.diracProba_comp_diracProbaEquiv_symm_eq_val
code: /-- The composition of `diracProbaEquiv.symm` and `diracProba` is the subtype inclusion. -/
lemma diracProba_comp_diracProbaEquiv_symm_eq_val [T0Space X] :
    diracProba ∘ (diracProbaEquiv (X := X)).symm = fun μ ↦ μ.val 
goal: MeasureTheory.diracProba ∘ ⇑MeasureTheory.diracProbaEquiv.symm = fun μ => ↑μ

full_name: MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto
code: lemma tendsto_diracProbaEquivSymm_iff_tendsto [T0Space X] [CompletelyRegularSpace X]
    {μ : range (diracProba (X := X))} (F : Filter (range (diracProba (X := X)))) :
    Tendsto diracProbaEquiv.symm F (𝓝 (diracProbaEquiv.symm μ)) ↔ Tendsto id F (𝓝 μ) 
goal: Filter.Tendsto (⇑MeasureTheory.diracProbaEquiv.symm) F (𝓝 (MeasureTheory.diracProbaEquiv.symm μ)) ↔
  Filter.Tendsto id F (𝓝 μ)

full_name: MeasureTheory.continuous_diracProbaEquiv
code: /-- In a T0 topological space, `diracProbaEquiv` is continuous. -/
lemma continuous_diracProbaEquiv [T0Space X] :
    Continuous (diracProbaEquiv (X := X)) 
goal: Continuous ⇑MeasureTheory.diracProbaEquiv

full_name: MeasureTheory.continuous_diracProbaEquivSymm
code: /-- In a completely regular T0 topological space, the inverse of `diracProbaEquiv` is continuous. -/
lemma continuous_diracProbaEquivSymm [T0Space X] [CompletelyRegularSpace X] :
    Continuous (diracProbaEquiv (X := X)).symm 
goal: Continuous ⇑MeasureTheory.diracProbaEquiv.symm

full_name: MeasureTheory.diracProbaHomeomorph
code: /-- In a completely regular T0 topological space `X`, `diracProbaEquiv` is a homeomorphism to
its image in `ProbabilityMeasure X`. -/
noncomputable def diracProbaHomeomorph [T0Space X] [CompletelyRegularSpace X]
    [MeasurableSpace X] [OpensMeasurableSpace X] : X ≃ₜ range (diracProba (X := X)) 
goal: X ≃ₜ ↑(Set.range MeasureTheory.diracProba)

full_name: MeasureTheory.embedding_diracProba
code: /-- If `X` is a completely regular T0 space with its Borel sigma algebra, then the mapping
that takes a point `x : X` to the delta-measure `diracProba x` is an embedding
`X → ProbabilityMeasure X`. -/
theorem embedding_diracProba [T0Space X] [CompletelyRegularSpace X]
    [MeasurableSpace X] [OpensMeasurableSpace X] : Embedding (fun (x : X) ↦ diracProba x) 
goal: Embedding fun x => MeasureTheory.diracProba x