Theorems · Theorem · measure theory

MeasureTheory.measure_integral_le_pos

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ}
  [MeasureTheory.IsProbabilityMeasure μ], MeasureTheory.Integrable f μ → 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}

First moment method. An integrable function is greater than its integral on a set of positive measure.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

0 results in Mathlib

Foundations

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

Assumes

MeasureTheory.IsProbabilityMeasure