Corollary 4

Tag 08M4, permanent. Cite it as [Bourbaki, Tag 08M4].

Integration, chapter III, § 1. Measures on a locally compact space

— In the space $\mathscr{M}(X; \mathbf{C})$ , the mapping $\mu \mapsto \|\mu\|$ is lower semi-continuous for the vague topology.

This is an immediate consequence of Corollary 2.

It should be noted that the mapping $\mu \mapsto |\mu|$ of $\mathscr{M}(X; \mathbf{C})$ into itself is not necessarily continuous for the vague topology (Exer. 9).

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

Cited by

Corollary 1, § 2

Cites

Corollary 2