Proposition 2

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

Integration, V, p. 3, § 1. Essential upper integral

Let $f, g, h$ be three elements of $\mathscr{F}_+$ ; if $g$ and $h$ are measurable, then

\int^\bullet f(g + h)\, d\mu = \int^\bullet fg\, d\mu + \int^\bullet fh\, d\mu.

One is immediately reduced to the proof of the analogous formula for the upper integral. Since $f(g + h) = fg + fh$ (with the convention that $0 \cdot (+\infty) = 0$ ), we have

\int^* f(g + h)\, d\mu \leq \int^* fg\, d\mu + \int^* fh\, d\mu;

it remains to establish the reverse inequality. Let $u$ be a lower semi-continuous function such that $u \geq f(g + h)$ . Set $v = \frac{u}{g + h}$ on the set where $g + h > 0$ , and $v = +\infty$ on the set where $g + h = 0$ ; then $v \geq f$ and $u \geq v(g + h)$ , whence

\int^* v(g + h)\, d\mu \leq \int^* u\, d\mu

and consequently, $v$ being measurable (Ch. IV, §5, No. 6, Cor. 4 of Th. 5),

\int^* fg\, d\mu + \int^* fh\, d\mu \leq \int^* vg\, d\mu + \int^* vh\, d\mu \\ = \int^* v(g + h)\, d\mu \leq \int^* u\, d\mu,

which implies the desired inequality since $u$ is arbitrary.

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

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Cites

IV, §5, No. 6, Cor. 4