File: Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean

full_name: measurableSet_ball
code: @[measurability]
theorem measurableSet_ball : MeasurableSet (Metric.ball x ε) 
goal: MeasurableSet (Metric.ball x ε)

full_name: measurableSet_closedBall
code: @[measurability]
theorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x ε) 
goal: MeasurableSet (Metric.closedBall x ε)

full_name: measurable_infDist
code: @[measurability]
theorem measurable_infDist {s : Set α} : Measurable fun x => infDist x s 
goal: Measurable fun x => Metric.infDist x s

full_name: Measurable.infDist
code: @[measurability, fun_prop]
theorem Measurable.infDist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infDist (f x) s 
goal: Measurable fun x => Metric.infDist (f x) s

full_name: measurable_infNndist
code: @[measurability]
theorem measurable_infNndist {s : Set α} : Measurable fun x => infNndist x s 
goal: Measurable fun x => Metric.infNndist x s

full_name: Measurable.infNndist
code: @[measurability, fun_prop]
theorem Measurable.infNndist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infNndist (f x) s 
goal: Measurable fun x => Metric.infNndist (f x) s

full_name: measurable_dist
code: @[measurability]
theorem measurable_dist : Measurable fun p : α × α => dist p.1 p.2 
goal: Measurable fun p => dist p.1 p.2

full_name: Measurable.dist
code: @[measurability, fun_prop]
theorem Measurable.dist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => dist (f b) (g b) 
goal: Measurable fun b => dist (f b) (g b)

full_name: measurable_nndist
code: @[measurability]
theorem measurable_nndist : Measurable fun p : α × α => nndist p.1 p.2 
goal: Measurable fun p => nndist p.1 p.2

full_name: Measurable.nndist
code: @[measurability, fun_prop]
theorem Measurable.nndist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => nndist (f b) (g b) 
goal: Measurable fun b => nndist (f b) (g b)

full_name: measurableSet_eball
code: @[measurability]
theorem measurableSet_eball : MeasurableSet (EMetric.ball x ε) 
goal: MeasurableSet (EMetric.ball x ε)

full_name: measurable_edist_right
code: @[measurability, fun_prop]
theorem measurable_edist_right : Measurable (edist x) 
goal: Measurable (edist x)

full_name: measurable_edist_left
code: @[measurability, fun_prop]
theorem measurable_edist_left : Measurable fun y => edist y x 
goal: Measurable fun y => edist y x

full_name: measurable_infEdist
code: @[measurability]
theorem measurable_infEdist {s : Set α} : Measurable fun x => infEdist x s 
goal: Measurable fun x => EMetric.infEdist x s

full_name: Measurable.infEdist
code: @[measurability, fun_prop]
theorem Measurable.infEdist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infEdist (f x) s 
goal: Measurable fun x => EMetric.infEdist (f x) s

full_name: tendsto_measure_cthickening
code: /-- If a set has a closed thickening with finite measure, then the measure of its `r`-closed
thickenings converges to the measure of its closure as `r` tends to `0`. -/
theorem tendsto_measure_cthickening {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) :
    Tendsto (fun r => μ (cthickening r s)) (𝓝 0) (𝓝 (μ (closure s))) 
goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ (closure s)))

full_name: tendsto_measure_cthickening_of_isClosed
code: /-- If a closed set has a closed thickening with finite measure, then the measure of its closed
`r`-thickenings converge to its measure as `r` tends to `0`. -/
theorem tendsto_measure_cthickening_of_isClosed {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) (h's : IsClosed s) :
    Tendsto (fun r => μ (cthickening r s)) (𝓝 0) (𝓝 (μ s)) 
goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s))

full_name: tendsto_measure_thickening
code: /-- If a set has a thickening with finite measure, then the measures of its `r`-thickenings
converge to the measure of its closure as `r > 0` tends to `0`. -/
theorem tendsto_measure_thickening {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (thickening R s) ≠ ∞) :
    Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ (closure s))) 
goal: Filter.Tendsto (fun r => μ (Metric.thickening r s)) (𝓝[>] 0) (𝓝 (μ (closure s)))

full_name: tendsto_measure_thickening_of_isClosed
code: /-- If a closed set has a thickening with finite measure, then the measure of its
`r`-thickenings converge to its measure as `r > 0` tends to `0`. -/
theorem tendsto_measure_thickening_of_isClosed {μ : Measure α} {s : Set α}
    (hs : ∃ R > 0, μ (thickening R s) ≠ ∞) (h's : IsClosed s) :
    Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ s)) 
goal: Filter.Tendsto (fun r => μ (Metric.thickening r s)) (𝓝[>] 0) (𝓝 (μ s))

full_name: measurable_edist
code: @[measurability]
theorem measurable_edist : Measurable fun p : α × α => edist p.1 p.2 
goal: Measurable fun p => edist p.1 p.2

full_name: Measurable.edist
code: @[measurability, fun_prop]
theorem Measurable.edist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => edist (f b) (g b) 
goal: Measurable fun b => edist (f b) (g b)

full_name: AEMeasurable.edist
code: @[measurability, fun_prop]
theorem AEMeasurable.edist {f g : β → α} {μ : Measure β} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun a => edist (f a) (g a)) μ 
goal: AEMeasurable (fun a => edist (f a) (g a)) μ

full_name: tendsto_measure_cthickening_of_isCompact
code: /-- Given a compact set in a proper space, the measure of its `r`-closed thickenings converges to
its measure as `r` tends to `0`. -/
theorem tendsto_measure_cthickening_of_isCompact [MetricSpace α] [MeasurableSpace α]
    [OpensMeasurableSpace α] [ProperSpace α] {μ : Measure α} [IsFiniteMeasureOnCompacts μ]
    {s : Set α} (hs : IsCompact s) :
    Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s)) 
goal: Filter.Tendsto (fun r => μ (Metric.cthickening r s)) (𝓝 0) (𝓝 (μ s))

full_name: exists_borelSpace_of_countablyGenerated_of_separatesPoints
code: /-- If a measurable space is countably generated and separates points, it arises as
the borel sets of some second countable t4 topology (i.e. a separable metrizable one). -/
theorem exists_borelSpace_of_countablyGenerated_of_separatesPoints (α : Type*)
    [m : MeasurableSpace α] [CountablyGenerated α] [SeparatesPoints α] :
    ∃ τ : TopologicalSpace α, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α 
goal: ∃ τ, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α

full_name: exists_opensMeasurableSpace_of_countablySeparated
code: /-- If a measurable space on `α` is countably generated and separates points, there is some
second countable t4 topology on `α` (i.e. a separable metrizable one) for which every
open set is measurable. -/
theorem exists_opensMeasurableSpace_of_countablySeparated (α : Type*)
    [m : MeasurableSpace α] [CountablySeparated α] :
    ∃ τ : TopologicalSpace α, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α 
goal: ∃ τ, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α

full_name: measurable_norm
code: @[fun_prop, measurability]
theorem measurable_norm : Measurable (norm : α → ℝ) 
goal: Measurable norm

full_name: Measurable.norm
code: @[measurability, fun_prop]
theorem Measurable.norm {f : β → α} (hf : Measurable f) : Measurable fun a => norm (f a) 
goal: Measurable fun a => ‖f a‖

full_name: AEMeasurable.norm
code: @[measurability, fun_prop]
theorem AEMeasurable.norm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => norm (f a)) μ 
goal: AEMeasurable (fun a => ‖f a‖) μ

full_name: measurable_nnnorm
code: @[measurability]
theorem measurable_nnnorm : Measurable (nnnorm : α → ℝ≥0) 
goal: Measurable nnnorm

full_name: Measurable.nnnorm
code: @[measurability, fun_prop]
theorem Measurable.nnnorm {f : β → α} (hf : Measurable f) : Measurable fun a => ‖f a‖₊ 
goal: Measurable fun a => ‖f a‖₊

full_name: AEMeasurable.nnnorm
code: @[measurability, fun_prop]
theorem AEMeasurable.nnnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => ‖f a‖₊) μ 
goal: AEMeasurable (fun a => ‖f a‖₊) μ

full_name: measurable_ennnorm
code: @[measurability]
theorem measurable_ennnorm : Measurable fun x : α => (‖x‖₊ : ℝ≥0∞) 
goal: Measurable fun x => ↑‖x‖₊

full_name: Measurable.ennnorm
code: @[measurability, fun_prop]
theorem Measurable.ennnorm {f : β → α} (hf : Measurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) 
goal: Measurable fun a => ↑‖f a‖₊

full_name: AEMeasurable.ennnorm
code: @[measurability, fun_prop]
theorem AEMeasurable.ennnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
    AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ 
goal: AEMeasurable (fun a => ↑‖f a‖₊) μ