arXiv:0806.1247v2 Announce Type: replace Abstract: Let $\mu$ be a finite non-atomic measure on a set $X$. A \emph{segment} for $\mu$ is an increasing family $u(t)$, $t\in[0,1]$, of measurable sets with $u(0)=\emptyset$, $u(1)=X$ and $\mu(u(t))=t\,\mu(X)$; by a theorem of Fryszkowski, finitely many measures $f\,d\mu$ admit a common segment. We study \emph{open segments}, i.e.\ segments consisting of open sets, when $X$ carries a topology. We show that weak outer regularity of $\mu$ is equivalent to the approximability of measurable sets by open sets, and that it holds for every finite Borel measure on a metrizable space. Our main result states that if $\mu$ is finite, non-atomic and weakly outer regular, then for all $f,g\in L^1(\mu)$ with $(\int f\,d\mu,\int g\,d\mu)\neq(0,0)$ there is a common open segment for $f\,d\mu$ and $g\,d\mu$; in particular, for every $f\in L^1(\mu)$ there is an open segment for $\mu$ which is simultaneously a segment for $f\,d\mu$. Relative versions allow one to prescribe an initial set. We also characterize open segments as upper semicontinuous functions with uniform distribution and show that on separable metrizable spaces a single measure admits such a function which is continuous.
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