Proposition 1

Tag 088Y, permanent. Cite it as [Bourbaki, Tag 088Y].

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

Let A be a set filtered by a filter $\mathfrak{F}$ , and $(f_\alpha)_{\alpha \in A}$ a family of regulated functions on a compact interval $I = [a, b]$ ; if the functions $f_\alpha$ converge uniformly on I to a (regulated) function f with respect to the filter $\mathfrak{F}$ , then

\lim_{\mathfrak{F}} \int_a^b f_\alpha(t) \, dt = \int_a^b f(t) \, dt.

Two corollaries to this proposition are important in applications:

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

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