Theorems · Theorem · measure theory

MeasureTheory.exists_notMem_null_le_lintegral

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ENNReal}
  [MeasureTheory.IsProbabilityMeasure μ], AEMeasurable f μ → μ N = 0 → ∃ x ∉ N, f x ≤ ∫⁻ (a : α), f a ∂μ

First moment method. The minimum of a measurable function is smaller than its integral, while avoiding a null set.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

0 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasure