lintegral_eq_zero_iff — Mathlib

English

If f is AEMeasurable, then lintegral μ f = 0 iff f =ᵐ μ 0.

Русский

Если f — AEMеас measurable, то линегральный интеграл μ f = 0 тогда и только тогда, когда f =ᵐ μ 0.

LaTeX

$$$\\text{AEMeasurable}(f, μ) \\Rightarrow (\\int f \\, d\\mu = 0 \\iff f =^\\ae_μ 0)$$$

Lean4

/-- The measurability assumption is necessary, otherwise there are counterexamples: for instance,
the conclusion fails if `f` is the characteristic function of a Vitali set. -/
@[simp]
theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
  lintegral_eq_zero_iff' hf.aemeasurable

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