There Is No Infinite - Dimensional Lebesgue Measure
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High Quality Content by WIKIPEDIA articles! In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite dimensional spaces: often, the abstract Wiener space construction is used. Alternatively, one may consider Lebesgue measure on finite-dimensional subspaces of the larger space and consider so-called prevalent and shy sets. Compact sets in Banach spaces may also carry natural measures: the Hilbert cube, for instance, carries the product Lebesgue measure. In a similar spirit, the c...
High Quality Content by WIKIPEDIA articles! In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite dimensional spaces: often, the abstract Wiener space construction is used. Alternatively, one may consider Lebesgue measure on finite-dimensional subspaces of the larger space and consider so-called prevalent and shy sets. Compact sets in Banach spaces may also carry natural measures: the Hilbert cube, for instance, carries the product Lebesgue measure. In a similar spirit, the compact topological group given by the Tychonoff product of infinitely many copies of the circle group is infinite-dimensional, and carries a Haar measure that is translation-invariant.
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