Proposition 3

— Let X be a locally compact subspace of T, and $\lambda$ a complex measure on X. The following properties are equivalent: a) the canonical injection $i : X \to T$ is $\lambda$-proper; b) for every compact subset K of T, $K \cap X$ is essentially $\lambda$-integrable; c) every point $t \in T$ admits a neighborhood V such that $V \cap X$ is essentially $\lambda$-integrable; d) there exists a measure $\theta$ on T such that $\theta_X = \lambda$. If these equivalent conditions are satisfied, we have, with notations as in d),

The injection $i$ being continuous, the equivalence of the properties a), b) and c) follows from Prop. 1 of §6, and the remark that follows it, applied to the positive measure $|\lambda|$. If $\lambda$ is induced on X by a measure $\theta$ on T, then $|\lambda| = |\theta|_X$ (formula (6)), consequently $|\lambda|(K \cap X) = |\theta|(K \cap X) \leq |\theta|(K) < +\infty$ (Prop. 1) for every compact subset K of T, so that d) implies b). Suppose that a) is satisfied, and let us show that $(i(\lambda))_X = \lambda$, which will imply d). Let g be an element of $\mathscr{H}(X; \mathbf{C})$; denoting by $g'$ the extension by 0 of g to T we have, by the definition of induced measure and then by Prop. 7 of §6, No. 4,

This completes the proof of the equivalence of the four properties. If $\lambda = \theta_X$ and $g \in \mathscr{H}(T; \mathbf{C})$, then

because $g \varphi_X$ is the extension by 0 of $g \circ i$ to T. This proves the second formula of (7).