# Complete measure

A measure $\mu$ on a $\sigma$- algebra $\Sigma$ for which $A \in \Sigma$ and $| \mu | ( A) = 0$ imply $E \in \Sigma$ for every $E \subset A$. Here $| \mu |$ is the total variation of $\mu$( $| \mu | = \mu$ for a positive measure).

Complete measures arise as follows (cf. [a1]). Let $X$ be a set, $\Sigma$ a $\sigma$- algebra of subsets of it and $\mu$ a positive measure on $\Sigma$. It may happen that some set $E \in \Sigma$ with $\mu ( E) = 0$ has a subset $N$ not belonging to $\Sigma$. It is natural, then, to define the measure $\mu$ on such a set $N$ as $\mu ( N) = 0$.
In general, let $\Sigma ^ {*}$ be the collection of all sets $N$ for which there exists sets $E , F \in \Sigma$ such that $E \subset N \subset F$, $\mu ( F - E ) = 0$. In this situation, define $\mu ( N) = 0$. Then $\Sigma ^ {*}$ is a $\sigma$- algebra and $\mu$ becomes a complete measure on it (this process is called completion). $( X , \Sigma ^ {*} , \mu )$ is then called a complete measure space.