Theorems · Theorem · measure theory

MeasureTheory.Measure.sum_restrict_le

∀ {α : Type u_2} {ι : Type u_6} {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ι → Set α} {M : ℕ},
  (∀ (i : ι), MeasurableSet (s i)) →
    (∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M) →
      (MeasureTheory.Measure.sum fun i => μ.restrict (s i)) ≤ M • μ.restrict (⋃ i, s i)

An upper bound on a sum of restrictions of a measure μ. This can be used to compare ∫ x ∈ X, f x ∂μ with ∑ i, ∫ x ∈ (s i), f x ∂μ, where s is a cover of X.

Defined in

Mathlib.MeasureTheory.Measure.Restrict

Cited by

0 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound