Theorems · Theorem · probability

ProbabilityTheory.absolutelyContinuous_of_posterior

∀ {Ω : Type u_1} {𝓧 : Type u_2} {mΩ : MeasurableSpace Ω} {m𝓧 : MeasurableSpace 𝓧} {κ : ProbabilityTheory.Kernel Ω 𝓧}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsFiniteMeasure μ] [inst_1 : ProbabilityTheory.IsFiniteKernel κ]
  [inst_2 : StandardBorelSpace Ω] [inst_3 : Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧],
  (∀ᵐ (b : 𝓧) ∂μ.bind ⇑κ, ((ProbabilityTheory.posterior κ μ) b).AbsolutelyContinuous μ) →
    ∀ᵐ (ω : Ω) ∂μ, (κ ω).AbsolutelyContinuous (μ.bind ⇑κ)

Defined in

Mathlib.Probability.Kernel.Posterior

Cited by

1 results in Mathlib

Foundations

Depth 336 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasureProbabilityTheory.IsFiniteKernelStandardBorelSpaceNonemptyMeasurableSpace.CountableOrCountablyGenerated