Remark 3

Tag 09PU, permanent. Cite it as [Bourbaki, Tag 09PU].

Integration, IX, p. 45, § 3. Measures and additive set functions

Let $(A_n)$ be any sequence of Borel sets of T. If I is countably additive, then $I(\bigcup_n A_n) \leq \sum_n I(A_n)$ . By the preceding remark, it suffices to establish this inequality for a finite sequence. One immediately reduces to the case of two sets $A_1$ and $A_2$ ; but the relation (4) implies that

I(A_1 \cup A_2) = I(A_1) + I(A_2 - (A_1 \cap A_2)) \leq I(A_1) + I(A_2).

The desired inequality then follows immediately.

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

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