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In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space. ==Motivation== The standard Gaussian measure γ''n'' on ''n''-dimensional Euclidean space R''n'' is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the ''n''-dimensional Lebesgue measure, denoted here ''dx''.) Instead, a measurable subset ''A'' has Gaussian measure : Here refers to the standard Euclidean dot product in R''n''. The Gaussian measure of the translation of ''A'' by a vector ''h'' ∈ R''n'' is : So under translation through ''h'', the Gaussian measure scales by the distribution function appearing in the last display: : The measure that associates to the set ''A'' the number γ''n''(''A''−''h'') is the pushforward measure, denoted (''T''''h'')∗(γ''n''). Here ''T''''h'' : R''n'' → R''n'' refers to the translation map: ''T''''h''(''x'') = ''x'' + ''h''.. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by : Abstract Wiener measure ''γ'' on a separable Banach space ''E'', where ''i'' : ''H'' → ''E'' is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace ''i''(''H'') ⊆ ''E''. 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「Cameron–Martin theorem」の詳細全文を読む スポンサード リンク
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