compactly supported continuous functions are dense in
for all compact sets .
is inner regular, meaning
is outer regular, meaning
We denote by the space of continuous functions with compact support.
Theroem - For every , is dense in (http://planetmath.org/LpSpace).
We begin by proving that for each with finite measure, the characteristic function can be approximated, in the norm, by functions in . Let . By of , we know there exist an open set and a compact set such that and
By the Urysohn’s lemma for locally compact Hausdorff spaces (http://planetmath.org/ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces), we know there is a function such that , and . Hence,
Thus, can be approximated in by functions in .
Now, it follows easily that any simple function , where each has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in we see that is also dense in .
|Title||compactly supported continuous functions are dense in|
|Date of creation||2013-03-22 18:38:53|
|Last modified on||2013-03-22 18:38:53|
|Last modified by||asteroid (17536)|
|Synonym||is dense in|