condExp_nonpos — Mathlib

English

Conditional expectation is homogeneous with real scalar: condExpIndL1Fin hm hs hμs (c • x) = c • condExpIndL1Fin hm hs hμs x.

Русский

Условное ожидание линейно по константе: condExpIndL1Fin hm hs hμs (c • x) = c • condExpIndL1Fin hm hs hμs x.

LaTeX

$$$condExpIndL1Fin hm hs hμs (c \cdot x) = c \cdot condExpIndL1Fin hm hs hμs x$$$

Lean4

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

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