Outer Measures Generated by a Countably Additive Measure on a Ring of Sets

Abstract

Let $\mathcal{R}$ be any ring of subsets of a set $X$ which is not an algebra and let $\mathcal{A}$ be the algebra generated by $\mathcal{R}$. Suppose that $\mu$ is a countably additive measure on $\mathcal{R}$ and that $\mu^*$ is the outer measure generated by $(\mu,\mathcal{R})$. If $X$ is a countable union of sets in $\mathcal{R}$, then there is a unique countably additive measure $\nu$ on $\mathcal{A}$ which extends $\mu$, and the outer measure generated by $(\nu,\mathcal{A})$ coincides with $\mu^*$. If $X$ is not a countable union of sets in $\mathcal{R}$, then there exists a family $\{ \mu_p : 0 \leq p \leq \infty \}$ of countably additive measures on $\mathcal{A}$ such that each $\mu_p$ agrees with $\mu$ on $\mathcal{R}$. For $0 \leq p \leq \infty$, let $\mu_p^*$ denote the outer measure generated by $(\mu_p, \mathcal{A})$. Then we have $\mu_0^* \leq \mu_p^* \leq \mu_q^* \leq \mu_\infty^* =\mu^*$ for $0< p < q < \infty$. Moreover, if $\mathcal{M}$ and $\mathcal{M}_p$, respectively, denotes the $\sigma$-algebra of $\mu^*$-measurable and $\mu_p^*$-measurable sets, then $\mathcal{M}_p = \mathcal{M}_1 \subset \mathcal{M}_0 = \mathcal{M}_\infty = \mathcal{M}$ for all positive real numbers $p$. As examples, we give countably additive measures on rings for which $\mathcal{M} = \mathcal{M}_1$ and $\mathcal{M} \neq \mathcal{M}_1$, respectively. By the outer measures generated by $\mu$ we shall mean the outer measures $\mu^*$ and $\mu_p^*$ $(0 \leq p \leq \infty)$.