Definition 1

The weak topology on $\mathscr{M}^b(T)$ associated with the preceding duality between $\mathscr{C}^b(T)$ and $\mathscr{M}^b(T)$ is called the topology of tight convergence (or the tight topology) on $\mathscr{M}^b(T)$.

The tight topology is Hausdorff, by the remarks preceding the definition. We shall often employ the adverb 'tightly' to mean 'in the sense of the tight topology'. Absent mention to the contrary, $\mathscr{M}^b(T)$ will be equipped with the tight topology throughout the rest of this section.

Every element of $\mathscr{C}^b(T)$ is a linear combination of elements of $\mathscr{C}_+^b(T)$. For a filter $\mathfrak{F}$ on $\mathscr{M}^b(T)$ to converge tightly to a bounded measure $\lambda$, it is necessary and sufficient that