Proposition 8

— Let X be a completely regular space, T a subspace of X, and i the canonical injection of T into X. Denote by W the set of bounded positive measures on X that are concentrated on T, equipped with the topology induced by $\mathscr{M}^b(X)$. Then the mapping $\mu \mapsto i(\mu)$ of $\mathscr{M}_+^b(T)$ into $\mathscr{M}^b(X)$ is a homeomorphism of $\mathscr{M}_+^b(T)$ onto W.

We denote again by i the mapping $\mu \mapsto i(\mu)$ of $\mathscr{M}_+^b(T)$ into $\mathscr{M}_+^b(X)$; i is injective (§2, No. 4, Prop. 8) and maps $\mathscr{M}_+^b(T)$ into W (§2, No. 3, Prop. 7). If $\lambda \in W$, then $\lambda = i(\lambda_T)$ (§2, No. 3, Prop. 7 b)). Consequently, i is a bijection of $\mathscr{M}_+^b(T)$ onto W, and the inverse bijection of i is the mapping $r : \lambda \mapsto \lambda_T$ on $W$. On the other hand, $i$ is continuous: for, if $f \in \mathscr{C}^b(X)$, then $\langle i(\mu), f \rangle = \langle \mu, f \circ i \rangle$, and $f \circ i$ belongs to $\mathscr{C}^b(T)$. Thus, everything comes down to showing that, for every measure $\mu \in W$ and every function $f \in \mathscr{C}^b_+(T)$, one has

or again

Let $f^\infty$ be the function on $X$ that coincides with $f$ on $T$ and with $+\infty$ on $X - T$, and let $f'$ and $f''$ be, respectively, the upper semi-continuous regularization of $f^0$ and the lower semi-continuous regularization of $f^\infty$ (GT, IV, §6, No. 2). The relations

immediately imply that $f'$ and $f''$ both coincide with $f$ and $f^0$ on $T$. Prop. 6 then yields

But one can replace $f'$ and $f''$ by $f^0$ in these two formulas, since the measures $\lambda$ and $\mu$ are carried by $T$; we have thus obtained the desired relation.

The statement of Prop. 8 is only valid for positive measures: the mapping $\mu \mapsto i(\mu)$ of $\mathscr{M}^b(T)$ into $\mathscr{M}^b(X)$ is injective and continuous, but is not in general a homeomorphism of $\mathscr{M}^b(T)$ onto its image. For example, take $X = \mathbf{R}$, $T = \mathbf{R} - \{0\}$; the measures $\lambda_t = \varepsilon_t - \varepsilon_{-t}$ ($t > 0$) converge tightly to 0 in $X$ as $t$ tends to 0, but do not converge tightly to 0 in $T$ (the characteristic function of $]0, +\infty[$ belongs to $\mathscr{C}^b(T)$) (cf. however the Cor. of Th. 1 of No. 5).