integral_nonneg_of_ae — Mathlib

English

If ν ≥ μ and f is integrable w.r.t. ν, and f is almost everywhere nonnegative (or f is ordered), then the integral with respect to μ is bounded by the integral with respect to ν, i.e., ∫ f dμ ≤ ∫ f dν.

Русский

Если ν больше или равно μ и f интегрируем по ν, тогда интеграл по μ не превосходит интеграл по ν.

LaTeX

$$$\\mu \\le ν \\Rightarrow 0 \\le f \\Rightarrow \\int f \\,dμ \\le \\int f \\,dν$ (при подходящих условиях интегрируемости).$$

Lean4

/-- The integral of a function which is nonnegative almost everywhere is nonnegative. -/
theorem integral_nonneg_of_ae {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ x, f x ∂μ :=
  integral_eq_setToFun f ▸
    setToFun_nonneg (dominatedFinMeasAdditive_weightedSMul μ) (fun s _ _ => weightedSMul_nonneg s) hf

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