Theorems · Theorem · measure theory

MeasureTheory.measure_laverage_le_pos

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

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

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

3 results in Mathlib

Foundations

Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound