Definition 3

*One calls premeasure on a topological space $T$ every mapping $w$ that associates, to every compact subset $K$ of $T$, a measure $w_K$ on $K$, and that has the following property:

*If $K$ and $L$ are compact subsets of $T$ such that $K \subset L$, the measure $(w_L)_K$ induced by $w_L$ on $K$ is equal to $w_K$.

The premeasure $w$ is said to be real (resp. positive) if all of the measures $w_K$ are real (resp. positive).*

Let $w$ and $w'$ be two premeasures on $T$, $t$ a complex number; the premeasures $w + w'$ and $tw$ are defined by the formulas $(w + w')_K = w_K + w'_K$, $(tw)_K = tw_K$ for every compact subset $K$ of $T$. The premeasures on $T$ obviously form a vector space, which is denoted $\mathscr{P}(T; \mathbf{C})$; the space of real premeasures will be denoted $\mathscr{P}(T; \mathbf{R})$, or more often $\mathscr{P}(T)$, and the convex cone of positive premeasures will be denoted $\mathscr{P}_+(T)$. Let $w$ be a premeasure; the mapping $K \mapsto |w_K|$ is then a premeasure on $T$ (Ch. IV, §5, No. 7, Lemma 3), which will be denoted $|w|$. If $w$ is real, one sets w^+ = \frac{1}{2}(|w| + w),\ w^- = \frac{1}{2}(|w| - w); these two premeasures being positive, one sees that every real premeasure is the difference of two positive premeasures. Clearly $(w^+)_K = (w_K)^+$, $(w^-)_K = (w_K)^-$ for every compact subset K of T.

The vector space $P(T)$ is ordered by the cone $P_+(T)$. It is clear that $w^+ = \sup(w, 0),\ w^- = \sup(-w, 0)$; consequently, $P(T)$ is lattice-ordered and $\sup(w, w') = w + (w' - w)^+,\ \inf(w, w') = w - (w' - w)^-$. Moreover, clearly

for every compact subset K of T.