Multiple Images and Local Times of Measurable Functions
Simeon M. Berman · Proceedings of the American Mathematical Society · 1986
Let $x(t)$, $0 \leq t \leq 1$, be a real-valued measurable function having a local time ${\alpha _{[0,t]}}(x)$ which is continuous in $t$, for almost all $x$. Then, for every integer $m \geq 2$, and every nonempty open subinterval $J \subset [0,1]$, there exist $m$ disjoint subintervals ${I_1}, \ldots ,{I_m} \subset J$ such that the intersection of the images of ${I_1}, \ldots ,{I_m}$ under the mapping $t \to x(t)$ has positive Lebesgue measure. The result applies to a large class of sample functions of stochastic processes, and also to multidimensional $t$ and $x( \cdot )$.