Exercise 1, § 2

Let T_1 and T_2 be two topological spaces, and f an injective continuous mapping of T_1 into T_2. Show that the mapping $\mu \mapsto f(\mu)$ is a bijection of $\mathscr{M}^b(T_1)$ onto the subspace of $\mathscr{M}^b(T_2)$ consisting of the measures carried by $f(T_1)$.