Corollary

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

Integration, chapter IV, § 5. Measurable functions and sets

— Every lower (resp. upper) semi-continuous function is measurable.

For, if $f$ is lower semi-continuous, then the set of $x \in \mathrm{X}$ such that $f(x) \leqslant a$ is closed for every $a \in \overline{\mathbf{R}}$ .

When F is metrizable and compact, Prop. 7 makes it possible to sharpen Th. 3 as follows:

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

Cited by

§ 5. Measurable functions and sets
Proposition 8, § 2
Exercise 17, § 1

Cites

Prop. 7
Th. 3