compactly supported continuous functions are dense in Lp

Let (X,,μ) be a measure spaceMathworldPlanetmath, where X is a locally compact Hausdorff spacePlanetmathPlanetmath, a σ-algebra ( that contains all compact subsets of X and μ a measure such that:

  • μ(K)< for all compact sets KX.

  • μ is inner regular, meaning μ(A)=sup{μ(K):KA,Kis compact}

  • μ is outer regular, meaning μ(A)=inf{μ(U):AU,UandUis open}

We denote by Cc(X) the space of continuous functionsMathworldPlanetmathPlanetmath X with compact support.

Theroem - For every 1p<, Cc(X) is dense in Lp(X) (

: It is clear that Cc(X) is indeed contained in Lp(X), where we identify each function in Cc(X) with its class in Lp(X).

We begin by proving that for each A with finite measure, the characteristic functionMathworldPlanetmathPlanetmathPlanetmath χA can be approximated, in the Lp norm, by functions in Cc(X). Let ϵ>0. By of μ, we know there exist an open set U and a compact set K such that KAU and


By the Urysohn’s lemma for locally compact Hausdorff spaces (, we know there is a function fCc(X) such that 0f1, f|K=1 and suppfU. Hence,


Thus, χA can be approximated in Lp by functions in Cc(X).

Now, it follows easily that any simple functionMathworldPlanetmathPlanetmath i=1nciχAi, where each Ai has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in Lp(X) we see that Cc(X) is also dense in Lp(X).

Title compactly supported continuous functions are dense in Lp
Canonical name CompactlySupportedContinuousFunctionsAreDenseInLp
Date of creation 2013-03-22 18:38:53
Last modified on 2013-03-22 18:38:53
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 54C35
Classification msc 46E30
Classification msc 28C15
Synonym Cc(X) is dense in Lp(X)