TODO #
Less explicit arguments to cond_eq_zero_of_meas_eq_zero
theorem
ProbabilityTheory.sum_meas_smul_cond_fiber'
{Ω : Type u_1}
{α : Type u_2}
{m : MeasurableSpace Ω}
[MeasurableSpace α]
[MeasurableSingletonClass α]
{X : Ω → α}
(hX : Measurable X)
[finX : FiniteRange X]
(μ : MeasureTheory.Measure Ω)
[MeasureTheory.IsFiniteMeasure μ]
:
((FiniteRange.toFinset X).sum fun (x : α) => μ (X ⁻¹' {x}) • ProbabilityTheory.cond 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.