Definition 5

— Let $w$ be a premeasure on a topological space $T$. One says that $w$ is a measure (resp. a bounded measure) if the encumbrance $|w|^\bullet$ is locally bounded (resp. bounded) (cf. No. 1, Def. 2).

The set of complex measures on $T$ is obviously a vector space (Remark 1), which will be denoted $\mathscr{M}(T; \mathbf{C})$. The space of real measures will be denoted $\mathscr{M}(T; \mathbf{R})$ or more often $\mathscr{M}(T)$, and the cone of positive measures will be denoted $\mathscr{M}_+(T)$.

If $w$ is a complex measure, its real part and its imaginary part are real measures. If $w$ is a real measure, $w^+$ and $w^-$ are positive measures. Every complex (resp. real) measure is thus a linear combination (resp. difference) of positive measures.