Corollary

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

Integration, chapter IX, § 6. Promeasures and measures on a locally convex space

— Let F be a barreled nuclear space, $\mathscr{T}$ a locally convex topology on $F'$ intermediate to $\mathscr{T}_s$ and $\mathscr{T}_c$ , $\mu$ a promeasure on $F'_\mathscr{T}$ , and $\Phi$ the Fourier transform of $\mu$ . For $\mu$ to be a measure, it is necessary and sufficient that $\Phi$ be continuous on F.

Necessity follows from Prop. 9 of No. 8 and sufficiency from Th. 2.

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

Cited by

Corollary, § 6

Cites

Prop. 9
Th. 2