Theorems · Theorem · measure theory

MeasureTheory.measure_le_laverage_pos

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal}
  [MeasureTheory.IsFiniteMeasure μ], μ ≠ 0 → AEMeasurable f μ → 0 < μ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}

First moment method. A measurable function is smaller than its mean on a set of positive measure.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

3 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure