lintegral_le_of_forall_fin_meas_le

English

For any measurable function f, ∫_{ {a} } f = f(a) μ({a}).

Русский

Для измеримой функции f выполняется ∫_{ {a} } f = f(a) μ({a}).

LaTeX

$$$\int_{\{a\}} f(x) \, d\mu = f(a) \mu\{a\}$$$

Lean4

/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure and the measure is σ-finite, then the integral over the whole space is bounded by that same
constant. -/
theorem lintegral_le_of_forall_fin_meas_le [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] (C : ℝ≥0∞) {f : α → ℝ≥0∞}
    (hf : ∀ s, MeasurableSet s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C :=
  have : SigmaFinite (μ.trim le_rfl) := by rwa [trim_eq_self]
  lintegral_le_of_forall_fin_meas_trim_le _ C hf

Похожие утверждения