Theorems · Theorem · functional analysis

Continuous.memLp_top_of_hasCompactSupport

∀ {E : Type u_4} [inst : NormedAddCommGroup E] {X : Type u_7} [inst_1 : TopologicalSpace X] [inst_2 : MeasurableSpace X]
  [OpensMeasurableSpace X] {f : X → E},
  Continuous f → HasCompactSupport f → ∀ (μ : MeasureTheory.Measure X), MeasureTheory.MemLp f ⊤ μ

A continuous function with compact support belongs to L^∞. See Continuous.memLp_of_hasCompactSupport for a version for L^p.

Defined in

Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Cited by

5 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupTopologicalSpaceMeasurableSpaceOpensMeasurableSpace