Proposition 5

Tag 096K, permanent. Cite it as [Bourbaki, Tag 096K].

Integration, chapter V, § 7. Integration with respect to an induced measure

— Let $(\lambda_\alpha)_{\alpha \in A}$ be an increasing directed family of positive measures on T, admitting a supremum λ, and let X be a locally compact subspace of T. The family of induced measures $\lambda_\alpha|X$ is then bounded above in $\mathscr{M}(X)$ , and

\sup_{\alpha \in A} (\lambda_\alpha|X) = \lambda|X.

In view of the identifications in the Scholium, this proposition is a special case of Prop. 5 of §5, No. 4.

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

Cited by

Nothing.

Cites

Prop. 5 of §5, No. 4