Theorems · Theorem · probability

ProbabilityTheory.condDistrib_apply_of_ne_zero

∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω]
  [inst_2 : Nonempty Ω] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst_3 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β}
  [MeasurableSingletonClass β],
  Measurable Y →
    ∀ (x : β),
      (MeasureTheory.Measure.map X μ) {x} ≠ 0 →
        ∀ (s : Set Ω),
          ((ProbabilityTheory.condDistrib Y X μ) x) s =
            ((MeasureTheory.Measure.map X μ) {x})⁻¹ * (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) ({x} ×ˢ s)

If the singleton {x} has non-zero mass for μ.map X, then for all s : Set Ω, condDistrib Y X μ x s = (μ.map X {x})⁻¹ * μ.map (fun a => (X a, Y a)) ({x} ×ˢ s) .

Defined in

Mathlib.Probability.Kernel.CondDistrib

Cited by

0 results in Mathlib

Foundations

Depth 280 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceStandardBorelSpaceNonemptyMeasureTheory.IsFiniteMeasureMeasurableSingletonClass