{"name":"continuous_pmf_apply","declaration":"theorem continuous_pmf_apply {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (fun s => (↑↑↑μ s).toNNReal) {i}"}
{"name":"probabilityMeasureEquivStdSimplex_symm_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_symm_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (p : ↑(stdSimplex ℝ X)) : ↑(probabilityMeasureEquivStdSimplex.symm p) =\n  Finset.sum Finset.univ fun i => ENNReal.ofReal (↑p i) • MeasureTheory.Measure.dirac i"}
{"name":"tendsto_lintegral_of_forall_of_finite","declaration":"theorem tendsto_lintegral_of_forall_of_finite {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] [Finite X] {ι : Type u_2} {L : Filter ι} (μs : ι → MeasureTheory.Measure X) (μ : MeasureTheory.Measure X) (f : BoundedContinuousFunction X NNReal) (h : ∀ (x : X), Filter.Tendsto (fun i => ↑↑(μs i) {x}) L (nhds (↑↑μ {x}))) : Filter.Tendsto (fun i => ∫⁻ (x : X), ↑(f x) ∂μs i) L (nhds (∫⁻ (x : X), ↑(f x) ∂μ))"}
{"name":"continuous_pmf_apply'","declaration":"theorem continuous_pmf_apply' {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (↑μ).real {i}"}
{"name":"probabilityMeasureEquivStdSimplex_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (μ : MeasureTheory.ProbabilityMeasure X) (i : X) : ↑(probabilityMeasureEquivStdSimplex μ) i = ↑((fun s => (↑↑↑μ s).toNNReal) {i})"}
{"name":"probabilityMeasureEquivStdSimplex","declaration":"/-- The canonical bijection between the set of probability measures on a fintype and the set of\nnonnegative functions on the points adding up to one. -/\ndef probabilityMeasureEquivStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] : MeasureTheory.ProbabilityMeasure X ≃ ↑(stdSimplex ℝ X)"}
{"name":"probabilityMeasureHomeoStdSimplex","declaration":"/-- The canonical homeomorphism between the space of probability measures on a finite space and the\nstandard simplex. -/\ndef probabilityMeasureHomeoStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] : MeasureTheory.ProbabilityMeasure X ≃ₜ ↑(stdSimplex ℝ X)"}
{"name":"instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"/-- This is still true when `X` is a metrizable compact space, but the proof requires Riesz\nrepresentation theorem.\nTODO: remove once the general version is proved. -/\ninstance instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : CompactSpace (MeasureTheory.ProbabilityMeasure X)"}
{"name":"instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"instance instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : SecondCountableTopology (MeasureTheory.ProbabilityMeasure X)"}