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 $\mu_p^*$ $(0 \leq p \leq \infty)$.
Real Anal. Exchange
27(1):
235-248
(2001/2002).
K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges, Academic Press, London,1983. MR751777 0516.28001 K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges, Academic Press, London,1983. MR751777 0516.28001