Theorems · Theorem · measure theory

MeasureTheory.exists_measurable_le_setLIntegral_eq_of_integrable

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ENNReal},
  ∫⁻ (a : α), f a ∂μ ≠ ⊤ →
    ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), MeasurableSet s → ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ

For any function f : α → ℝ≥0∞, there exists a measurable function g ≤ f with the same integral over any measurable set.

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Cited by

0 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace