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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Multiple images and local times of measurable functions


Author: Simeon M. Berman
Journal: Proc. Amer. Math. Soc. 97 (1986), 328-330
MSC: Primary 60J55; Secondary 60G17
DOI: https://doi.org/10.1090/S0002-9939-1986-0835892-8
MathSciNet review: 835892
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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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DOI: https://doi.org/10.1090/S0002-9939-1986-0835892-8
Keywords: Multiple image, local time
Article copyright: © Copyright 1986 American Mathematical Society