In search of a Lebesgue density theorem for R^\infty

Details In search of a Lebesgue density theorem for R^\infty In search of a Lebesgue density theorem for R^\infty

2008-10-27

arxiv.org/abs/0810.4894v1

Mathematics

Classical Analysis and ODEs

70 + vi pages, MSc research thesis defended on 22-Oct-2008 at the University of Ottawa, Canada (supervisor Vladimir Pestov)

Scientific paper

We look at a measure, $\lambda^\infty$, on the infinite-dimensional space, ${\mathbb R}^\infty$, for which we attempt to put forth an analogue of the Lebesgue density theorem. Although this measure allows us to find partial results, for example for continuous functions, we prove that it is impossible to give an analogous theorem in full generality. In particular, we proved that the Lebesgue density of probability density functions on ${\mathbb R}^\infty$ is zero almost everywhere.

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