Premeasure

Remarks

Example: on the ring of finite unions of half-open intervals $(a, b]$ in $\R$, the total length is a premeasure; countable additivity is checked with a compactness argument on closed subintervals. Non-example: on the ring of finite and cofinite subsets of $\N$, put $\mu_0(A) = 0$ for finite $A$ and $\mu_0(A) = 1$ for cofinite $A$. It is finitely additive, but $\N = \bigcup_n \set{n}$ gives $1 \ne \sum_n 0$, so clause 2 fails. A premeasure is monotone on $\mathcal{R}$, and countably subadditive for sequences in $\mathcal{R}$ whose union lies in $\mathcal{R}$. The same definition is used for a semiring of sets instead of a ring. A measure is a premeasure whose domain is a $\sigma$-algebra.