GAUSSIAN MEASURES IN FUNCTION SPACE
GAUSSIAN MEASURES IN FUNCTION SPACE
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函数空间中的高斯测度
DOI：
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发表时间：
1966
期刊：
影响因子：
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通讯作者：
L. A. Shepp
中科院分区：
文献类型：
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作者：
L. A. Shepp;L. A. Shepp
两个高斯测度要么是相互奇异的，要么是等价的。这种二分法首先由费尔德曼和哈耶克（独立）发现。我们给出了一个简单的，几乎正式的，证明这一结果的基础上，研究的某对泛函的两个措施。此外，我们表明，两个高斯措施与零均值和光滑Polya型协方差（在一个区间上）是等价的，当且仅当右手斜率的协方差在零是相等的。空间（Ω，&）上的H和J泛函 * 两个概率测度μ0和μx称为相互奇异（μ0 _L μd），如果存在一个集合μ e &，其中μ 0（B）- 0且μλ{Ω - B）= 0。如果这些测度具有相同的零点集，则称它们相互等价（μ0 ~ μ ±），即，μo（B）= 0当且仅当μ^B）= 0。设μ = μ0 + μx，我们可以定义Radon-Nikodym导数，
Two Gaussian measures are either mutually singular or equivalent. This dichotomy was first discovered by Feldman and Hajek (independently). We give a simple, almost formal, proof of this result, based on the study of a certain pair of functionals of the two measures. In addition we show that two Gaussian measures with zero means and smooth Polya-type covariances (on an interval) are equivalent if and only if the right-hand slopes of the covariances at zero are equal. The H and J functionals* Two probability measures μ0 and μx on a space (Ω, &) are called mutually singular (μ0 _L μd if there is a set ΰe & for which μo(B) — 0 and μλ{Ω — B) = 0. The measures are called mutually equivalent (μ0 ~ μ ±) if they have the same zero sets, i.e., μo(B) = 0 if and only if μ^B) = 0. Setting μ = μ0 + μx we may define the Radon-Nikodym derivatives,