Remark 4

Tag 08R6, permanent. Cite it as [Bourbaki, Tag 08R6].

Integration, IV, p. 25, § 3. LP spaces

If $f$ belongs to $\mathscr{L}_F^p$ , it is not always possible to find a sequence $(f_n)$ of continuous functions with compact support such that the sequence $(f_n(x))$ converges everywhere in $X$ to a function equal to $f(x)$ almost everywhere (§4, Exer. 4 c)).

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

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