condExp_nonneg — Mathlib

English

A paired additivity property for conditional expectations on finite-measure indicators.

Русский

Свойство аддитивности CondExpIndL1Fin для суммы

LaTeX

$$$condExpIndL1Fin hm hs hμs (x+y) = condExpIndL1Fin hm hs hμs x + condExpIndL1Fin hm hs hμs y$$$

Lean4

theorem condExp_nonneg (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] :=
  by
  by_cases hfint : Integrable f μ
  · rw [(condExp_zero.symm : (0 : α → E) = μ[0|m])]
    exact condExp_mono (integrable_zero _ _ _) hfint hf
  · rw [condExp_of_not_integrable hfint]

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