Multiple images and local times of measurable functions

Berman, Simeon M.

doi:10.1090/S0002-9939-1986-0835892-8

Multiple images and local times of measurable functions
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by Simeon M. Berman PDF

Proc. Amer. Math. Soc. 97 (1986), 328-330 Request permission

Abstract:

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 $x( \cdot )$.

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