Theorems · Theorem · measure theory

MeasureTheory.lintegral_le_of_forall_fin_meas_le

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (C : ENNReal)
  {f : α → ENNReal}, (∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) → ∫⁻ (x : α), f x ∂μ ≤ C

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.

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Cited by

0 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasureTheory.SigmaFinite