Theorems · Theorem · functional analysis

MeasureTheory.MemLp.mono_exponent_of_measure_support_ne_top

∀ {α : Type u_1} {ε' : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace ε']
  [inst_1 : ESeminormedAddMonoid ε'] {p q : ENNReal} {f : α → ε'},
  MeasureTheory.MemLp f q μ → ∀ {s : Set α}, (∀ x ∉ s, f x = 0) → μ s ≠ ⊤ → p ≤ q → MeasureTheory.MemLp f p μ

If a function is supported on a finite-measure set and belongs to ℒ^p, then it belongs to ℒ^q for any q ≤ p.

Defined in

Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Cited by

4 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceESeminormedAddMonoid