Theorems · Theorem · probability

ProbabilityTheory.Kernel.IndepSets.indep

∀ {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m1 m2 m : MeasurableSpace Ω}
  {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsZeroOrMarkovKernel κ]
  {p1 p2 : Set (Set Ω)},
  m1 ≤ m →
    m2 ≤ m →
      IsPiSystem p1 →
        IsPiSystem p2 →
          m1 = MeasurableSpace.generateFrom p1 →
            m2 = MeasurableSpace.generateFrom p2 →
              ProbabilityTheory.Kernel.IndepSets p1 p2 κ μ → ProbabilityTheory.Kernel.Indep m1 m2 κ μ

The measurable space structures generated by independent pi-systems are independent.

Defined in

Mathlib.Probability.Independence.Kernel.Indep

Cited by

9 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsZeroOrMarkovKernel