Proposition 21

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

Integration, IV, p. 85, § 5. Measurable functions and sets

Let $H$ be a subset of $\mathscr{L}_F^p(X, \mu)$ that is equi-integrable of order $p$ . On $H$ , the uniform structure of convergence in measure is equal to the uniform structure induced by that of $\mathscr{L}_F^p(X, \mu)$ .

Let $\varepsilon > 0$ . There exist $\delta$ and $K$ with the properties (i) and (ii) of Def. 10. Let $f, g$ in $H$ be such that

|f(x) - g(x)| \leq \left( \frac{\varepsilon}{|\mu|(K)} \right)^{1/p}

for $x \in K$ , except on a set $M$ of measure $\leq \delta$ . Then

\left( \int_{X-K} |f-g|^p d|\mu| \right)^{1/p} \leq \left( \int_{X-K} |f|^p d|\mu| \right)^{1/p} + \left( \int_{X-K} |g|^p d|\mu| \right)^{1/p}

\leq 2\varepsilon^{1/p}

and similarly

\left( \int_M |f-g|^p d|\mu| \right)^{1/p} \leq 2\varepsilon^{1/p},

therefore

\int |f-g|^p d|\mu| = \int_{X-K} |f-g|^p d|\mu| + \int_M |f-g|^p d|\mu| + \int_{K-M} |f-g|^p d|\mu|

\leq 2^p \varepsilon + 2^p \varepsilon + \frac{\varepsilon}{|\mu|(K)} |\mu|(K-M) \leq (2^{p+1} + 1)\varepsilon.

Thus, the uniform structure of convergence in measure on $H$ is finer than the uniform structure induced by that of $\mathscr{L}_F^p(X, \mu)$ . It then suffices to apply Prop. 20.

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

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Remark, § 5

Proposition 21

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