Proposition 15

Tag 09O3, permanent. Cite it as [Bourbaki, Tag 09O3].

Integration, chapter IX, § 1. Premeasures and measures on a topological space

The set of functions f with values in F, such that Supp(f) is compact and such that the restriction of f to Supp(f) is continuous, is dense in $\overline{\mathscr{L}}_F^p(\mu)$ and in $\mathscr{L}_F^p(\mu)$ , for $1 \leq p < +\infty$ .

Note that these functions are not continuous functions on T with compact support.

Let us pass to the definition of the integral.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 3, § 5

Cites

Nothing.