Young's functional with Lebesgue-Stieltjes integrals

摘要: For non-decreasing real functions $f$ and $g$, we consider the functional $ T(f,g ; I,J)=\int_{I} f(x)\di g(x) + \int_J g(x)\di f(x)$, where $I$ $J$ are intervals with $J\subseteq I$. In particular case $I=[a,t]$, $J=[a,s]$, $s\leq t$ $g(x)=x$, this reduces to expression in classical Young's inequality. We survey some properties of Lebesgue-Stieltjes interals present a new simple proof for change variables. Further, formulate version inequality respect arbitrary positive finite measure on line including purely discrete case, discuss an application related medians probability distributions summation formula that involves values function its inverse at integers.