Theorems · Theorem · probability

ProbabilityTheory.iIndep_iff

∀ {Ω : Type u_1} {ι : Type u_2} (m : ι → MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω),
  ProbabilityTheory.iIndep m μ ↔
    ∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, MeasurableSet (f i)) → μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)

Defined in: Mathlib.Probability.Independence.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites11

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Finsetstatement and proof · cited by 13,712
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement · cited by 9,879
Set.ofPredproof · cited by 6,101
MeasurableSetstatement and proof · cited by 3,075
Finset.prodstatement and proof · cited by 2,356
Set.iInterstatement and proof · cited by 1,084
ProbabilityTheory.iIndepstatement · cited by 30

Cited by1

Results whose statement or proof uses this declaration.

ProbabilityTheory.iIndep.meas_biInterproof · cited by 2