Corollary 2

Tag 0890, permanent. Cite it as [Bourbaki, Tag 0890].

Functions of a Real Variable, chapter II, § 3. DERIVATIVES AND INTEGRALS OF FUNCTIONS DEPENDING ON A PARAMETER

Let $A$ be a subset of a topological space $F$ , and $f$ a map from $I \times A$ into a complete normed space $E$ over $\mathbf{R}$ , such that, for each $\alpha \in A$ , the function $x \mapsto f(x, \alpha)$ is regulated on $I$ . If the functions $x \mapsto f(x, \alpha)$ converge uniformly on $I$ to a (regulated) function $x \mapsto g(x)$ , as $\alpha$ tends to a point $\alpha_0 \in \overline{A}$ while remaining in $A$ , then one has

\lim_{\alpha \to \alpha_0,\ \alpha \in A} \int_a^b f(x, \alpha)\, dx = \int_a^b g(x)\, dx.

In particular:

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

Cited by

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Cites

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