Theorems · Theorem · probability

ProbabilityTheory.sum_meas_smul_cond_fiber

∀ {Ω : Type u_1} {α : Type u_3} {m : MeasurableSpace Ω} [inst : Fintype α] [inst_1 : MeasurableSpace α]
  [DiscreteMeasurableSpace α] {X : Ω → α},
  Measurable X →
    ∀ (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ], ∑ x, μ (X ⁻¹' {x}) • μ[|X ⁻¹' {x}] = μ

The law of total probability for a random variable taking finitely many values: a measure μ can be expressed as a linear combination of its conditional measures μ[|X ← x] on fibers of a random variable X valued in a fintype.

Defined in

Mathlib.Probability.ConditionalProbability

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeMeasurableSpaceDiscreteMeasurableSpaceMeasureTheory.IsFiniteMeasure