Theorems · Theorem · measure theory

MeasureTheory.exists_notMem_null_le_average

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ}
  [MeasureTheory.IsFiniteMeasure μ], μ ≠ 0 → MeasureTheory.Integrable f μ → μ N = 0 → ∃ x ∉ N, f x ≤ ⨍ (a : α), f a ∂μ

First moment method. The minimum of an integrable function is smaller than its mean, while avoiding a null set.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

2 results in Mathlib

Foundations

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

Assumes

MeasureTheory.IsFiniteMeasure