expect_nonneg — Mathlib

English

If a function m is subadditive and respects rational scaling, then for a finite nonempty s and f valued in the domain, m of the average is bounded by the average of m(f(i)).

Русский

Если m субаддитивна и т. п., то m среднее по s ≤ среднее по i∈s m(f(i)).

LaTeX

$$$s\\neq \\varnothing \\Rightarrow m(\\mathbb{E}_{i\\in s} f(i)) \\le \\mathbb{E}_{i\\in s} m(f(i))$$$

Lean4

theorem expect_nonneg (hf : ∀ i ∈ s, 0 ≤ f i) : 0 ≤ 𝔼 i ∈ s, f i :=
  smul_nonneg (by positivity) <| sum_nonneg hf

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