condExp_congr_ae_trim — Mathlib

English

For integrable f under mh ≤ m0 and sigma-finite μ, μ[f|m] equals condExpL1CLM E hm μ (hf.toL1 f) almost everywhere.

Русский

При интегрируемом f и mh ≤ m0 и сигма-конечной μ, μ[f|m] равно condExpL1CLM почти наверху.

LaTeX

$$$$\\mu[f|m] =_{\\text{ae}} condExpL1CLM E hm μ (hf.toL1 f).$$$$

Lean4

theorem condExp_congr_ae_trim (hm : m ≤ m₀) (hfg : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ.trim hm] μ[g|m] :=
  StronglyMeasurable.ae_eq_trim_of_stronglyMeasurable hm stronglyMeasurable_condExp stronglyMeasurable_condExp
    (condExp_congr_ae hfg)

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