Proposition 7

Tag 09IT, permanent. Cite it as [Bourbaki, Tag 09IT].

Integration, VIII, p. 13, § 2. Linear representations of groups

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves stable $\mathscr{M}^1(X)$ and, if $\mathscr{M}^1(X)$ is equipped with the topology of uniform convergence in the compact subsets of $\mathscr{K}(X)$ , then the linear representation $\gamma_\chi$ of $G$ in $\mathscr{M}^1(X)$ is continuous. These propositions result from Props. 3, 4, 5.

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

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