Theorems · Theorem · measure theory

MeasureTheory.measure_setLAverage_le_pos

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

2 results in Mathlib

Foundations

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