Proposition 10

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

— For every measure $\mu$ on $X$ ,

(22)

\|\mu\| = \sup_{0 \leq f \leq 1,\ f \in \mathscr{H}(X; \mathbf{R})} |\mu|(f).

For, taking into account the formula (12) that defines the absolute value of a measure, the second member of (22) may be written

\sup_{0 \leq f \leq 1,\ f \in \mathscr{H}(X; \mathbf{R})} \left( \sup_{|g| \leq f,\ g \in \mathscr{H}(X; \mathbf{C})} |\mu(g)| \right) = \sup_{\|g\| \leq 1,\ g \in \mathscr{H}(X; \mathbf{C})} |\mu(g)|.

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

Cited by

Nothing.