PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure.jsonl

{"name":"lintegral_indicatorBCF","declaration":"theorem lintegral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫⁻ (x : α), ENNReal.ofReal ((indicatorBCF s_clopen) x) ∂μ = ↑↑μ s"}
{"name":"integral_indicatorBCF","declaration":"theorem integral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫ (x : α), (indicatorBCF s_clopen) x ∂μ = (↑↑μ s).toReal"}
{"name":"continuous_integral_finiteMeasure","declaration":"theorem continuous_integral_finiteMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"}
{"name":"indicatorBCF_apply","declaration":"theorem indicatorBCF_apply {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) (x : α) : (indicatorBCF s_clopen) x = Set.indicator s (fun x => 1) x"}
{"name":"indicatorBCF","declaration":"/-- The indicator function of a clopen set, as a bounded continuous function. -/\ndef indicatorBCF {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) : BoundedContinuousFunction α ℝ"}
{"name":"continuous_integral_probabilityMeasure","declaration":"theorem continuous_integral_probabilityMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"}