Remark 1

If $T$ is locally compact, the tight topology is finer than the topology induced on $\mathscr{M}^b(T)$ by the vague topology, and these two topologies coincide only when $T$ is compact. For, if $T$ is not compact, the mapping $t \mapsto \varepsilon_t$ converges vaguely to 0 with respect to the filter of complements of relatively compact subsets of $T$, but does not converge tightly to 0, because the function 1 belongs to $\mathscr{C}^b(T)$ (for the relations between vague convergence and tight convergence, see Prop. 9).

(1) This relation was only established above (§ 2, No. 1, Prop. 1) in the case that $f$ and $\nu$ are positive. The extension to the present situation, where $f$ and $\nu$ are complex and bounded, is immediate by linearity.