Definition (Pre-measure)

A pre-measure is a mapping $\lambda:\mathscr{A}\to[0,\infty]$ where $\mathscr{A}\subseteq 2^{X}$ is a Algebra of subsets of $X$ such that

1. $\lambda(\emptyset)=0$
2. Countable Additivity: $$\lambda\left(\bigsqcup_{j=1}^{\infty}A_{j}\right)=\sum\limits_{j=1}^{\infty}\lambda(A_{j})$$for $A_{j}\in\mathscr{A}$, $A_{i}\cap A_{j}=\emptyset$ for $i\not=j$.