Mathematics – Differential Geometry
Scientific paper
2010-12-11
Mathematics
Differential Geometry
typo corrected, section 1.3. enlarged, references added
Scientific paper
The goal of this article is to define the mean value of a function defined on an infinite product of measured spaces with infinite measure. As a preliminary approach, the mean value of a map defined on a $\sigma-$finite Radon measure $\mu$ with respect to a sequence of measurable sets called renormalization sequence. If $\mu$ is a probability measure, we recover the expectation value of a random variable. We also show that in many standard cases, if the measure is not finite, we get a linear extension of the limit at infinity. We investigate basic properties, especially invariance properties and formulas for changing the measure. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (but principally on a Hilbert space) is defined. Even if the renormalization procedure is fixed, it depends on a chosen orthogonal sequence. Once this sequence is fixed, the mean value is invariant through scaling and translation. We finally remark a restriction invariance, which is a fundamental difference with measures defined on Hilbert spaces.
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