Regular dependence of total variation on parameters (Q1769725)

If \(X\) is an interval, \(Y\) -- a metric space, \(T\) -- a set of parameters, and \(f: T\times X\to Y\) a function, then it can happen that \(f\) is measurable with respect to some \(\sigma\)-algebra while the function \(v:T\to X\), defined by \(v(t)\) equals to the total variation of \(f(t,\cdot)\), is not measurable. The paper presents several theorems giving the sufficient condition for regular (measurable, continuous) behaviour of \(v\).