Regular dependence of total variation on parameters.

Abstract

Let $X$ be an interval, $Y$ a metric space, $T$ a set of parameters, and $f:T\times X\to Y$ a function. For given $t\in T$ denote by $v(t)$ the total variation of $f(t,\cdot )$ on $X$. We look for sufficient conditions for regular (measurable, continuous, etc.) dependence of $v$ on $t$.