Corollary 2

Tag 087Q, permanent. Cite it as [Bourbaki, Tag 087Q].

Functions of a Real Variable, chapter II, § 1. PRIMITIVES AND INTEGRALS

*Let $f$ and $g$ be two real regulated functions on I, such that $g(x) \geq 0$ at the points where $g$ is continuous; if $m$ and $M$ are the greatest lower bound and least upper bound of $f$ over the set of points where $f$ is continuous, then

\frac{m}{b-a} \int_a^b g(t)\,dt \leq \frac{1}{b-a} \int_a^b f(t)g(t)\,dt \leq \frac{M}{b-a} \int_a^b g(t)\,dt.

(14)

The first two terms (resp. the two last) are unequal unless $g(x)(f(x) - m) = 0$ (resp. $g(x)(f(x) - M) = 0$ ) at every point where $f$ and $g$ are continuous.

For vector functions the mean value theorem (I, p. 15, th. 2) yields the following proposition:

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

Cited by

Nothing.

Cites

I, p. 15