Remark 2

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

Integration, IX, p. 44, § 3. Measures and additive set functions

The set of conditions 2) and 3) may be replaced, in the statement of Theorem 1, by the following condition: If $K$ and $L$ are compact subsets of $T$ , then

I(K \cup L) + I(K \cap L) = I(K) + I(L).

Indeed, this condition implies 2) and 3), and on the other hand

\mu^*(K \cup L) = \mu^*(K \cap L) = \mu^*(K) + \mu^*(L)

for every measure

\mu

, by virtue of the relation

\varphi_{K \cup L} + \varphi_{K \cap L} = \varphi_K + \varphi_L

between the characteristic functions.

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

Cited by

Nothing.

Cites

Theorem 1