Bourbaki

Proposition 8

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

Integration, chapter IV, § 1. Upper integral of a positive function

— Let (G_ι)_{ι \in I} be any family of open sets; then

μ∗(⋃ι∈IGι)≤∑ι∈Iμ∗(Gι). \mu^*\left(\bigcup_{\iota \in I} G_\iota\right) \leq \sum_{\iota \in I} \mu^*(G_\iota).

Moreover, if the G_ι are pairwise disjoint then

μ∗(⋃ι∈IGι)=∑ι∈Iμ∗(Gι). \mu^*\left(\bigcup_{\iota \in I} G_\iota\right) = \sum_{\iota \in I} \mu^*(G_\iota).

For, if G = ⋃ι∈IGι\bigcup_{\iota \in I} G_\iota then φ_G = sup_{ι \in I} φ_{G_ι} ≤ ∑ι∈IφGι\sum_{\iota \in I} \varphi_{G_\iota}; when the G_ι are pairwise disjoint, φ_G = ∑ι∈IφGι\sum_{\iota \in I} \varphi_{G_\iota}; the proposition is therefore a consequence of Props. 2 and 3.

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

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Cites

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Transcribed from the printed volume. Not translated.