O-segments on topological measure spaces
Let $μ$ be a finite non-atomic measure on a set $X$. A \emph{segment} for $μ$ is an increasing family $u(t)$, $t\in[0,1]$, of measurable sets with $u(0)=\emptyset$, $u(1)=X$ and $μ(u(t))=t\,μ(X)$; by a theorem of Fryszkowski, finitely many measures $f\,dμ$ 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 $μ$ 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 $μ$ is finite, non-atomic and weakly outer regular, then for all $f,g\in L^1(μ)$ with $(\int f\,dμ,\int g\,dμ)\neq(0,0)$ there is a common open segment for $f\,dμ$ and $g\,dμ$; in particular, for every $f\in L^1(μ)$ there is an open segment for $μ$ which is simultaneously a segment for $f\,dμ$. 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.