Nonincrease almost everywhere of certain measurable functions with applications to stochastic processes

Abstract:

Let $x(t)$, $0 \leqslant t \leqslant 1$, be a real valued measurable function having a local time ${\alpha _{[0,t]}}(x)$, $0 \leqslant t \leqslant 1$. If the latter is continuous in $t$ for almost all $x$, then almost every $t$ is not a point of increase of the function $x( \cdot )$.