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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. ==Motivation== It can be shown that Lebesgue measure ''λ''''n'' on Euclidean space R''n'' is locally finite, strictly positive and translation-invariant, explicitly: * every point ''x'' in R''n'' has an open neighbourhood ''N''''x'' with finite measure ''λ''''n''(''N''''x'') < +∞; * every non-empty open subset ''U'' of R''n'' has positive measure ''λ''''n''(''U'') > 0; and * if ''A'' is any Lebesgue-measurable subset of R''n'', ''T''''h'' : R''n'' → R''n'', ''T''''h''(''x'') = ''x'' + ''h'', denotes the translation map, and (''T''''h'')∗(''λ''''n'') denotes the push forward, then (''T''''h'')∗(''λ''''n'')(''A'') = ''λ''''n''(''A''). Geometrically speaking, these three properties make Lebesgue measure very nice to work with. When we consider an infinite-dimensional space such as an ''Lp'' space or the space of continuous paths in Euclidean space, it would be nice to have a similarly nice measure to work with. Unfortunately, this is not possible. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Infinite-dimensional Lebesgue measure」の詳細全文を読む スポンサード リンク
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