Remark 1
Tag 09NU, permanent. Cite it as [Bourbaki, Tag 09NU].
Integration, chapter IX, § 1. Premeasures and measures on a topological space
On a Lindelöf space T (TG, IX, Appendix I, Def. 1),¹ and in particular on a Souslin space (ibid., Cor. of Prop. 1), every measure is moderated. For, the open sets of finite measure form a covering of T, from which one can extract a countable covering of T.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.