condVar_of_stronglyMeasurable — Mathlib

English

Under aestronglyMeasurable assumptions, Var[X; μ | m] equals (X − μ[X|m])^2 almost surely.

Русский

При AEStronglyMeasurable предположениях Var[X; μ | m] равно (X − μ[X|m])^2 почти surely.

LaTeX

$$$ Var[X; μ | m] =^{a.e.}_{μ} (X - μ[X|m])^2 $$$

Lean4

theorem condVar_of_stronglyMeasurable [SigmaFinite (μ.trim hm)] (hX : StronglyMeasurable[m] X)
    (hXint : Integrable ((X - μ[X|m]) ^ 2) μ) : Var[X; μ | m] = fun ω ↦ (X ω - (μ[X|m]) ω) ^ 2 :=
  condExp_of_stronglyMeasurable _ ((hX.sub stronglyMeasurable_condExp).pow _) hXint

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