{"name":"MeasureTheory.Measure.pi_pi_finset","declaration":"theorem MeasureTheory.Measure.pi_pi_finset {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Finset ι) (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi (↑t) s) = Finset.prod t fun i => ↑↑(μ i) (s i)"}
{"name":"MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi","declaration":"instance MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.pi μ)"}
{"name":"MeasureTheory.Measure.pi_eval_preimage","declaration":"theorem MeasureTheory.Measure.pi_eval_preimage {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) (s : Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Function.eval i ⁻¹' s) = ↑↑(μ i) s"}
{"name":"MeasureTheory.Measure.map_eval_pi","declaration":"theorem MeasureTheory.Measure.map_eval_pi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) : MeasureTheory.Measure.map (Function.eval i) (MeasureTheory.Measure.pi μ) = μ i"}
{"name":"MeasureTheory.Measure.pi_pi_set","declaration":"theorem MeasureTheory.Measure.pi_pi_set {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Set ι) [DecidablePred fun x => x ∈ t] (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi t s) =\n  Finset.prod (Finset.filter (fun x => x ∈ t) Finset.univ) fun i => ↑↑(μ i) (s i)"}