Structure theorem for Gaussian measures

<h2 id="statement-of-the-theorem">Statement of the theorem</h2>
Let γ be a strictly positive Gaussian measure on a separable Banach space (E, || ||). Then there exists a <a href="/facts/Separable_Hilbert_space/aDFMh4lV">separable Hilbert space</a> (H, ⟨ , ⟩) and a map i : H → E such that i : H → E is an abstract Wiener space with γ = i∗(γH), where γH is the <a href="/facts/Canonical_form/K2jhB44k">canonical</a> Gaussian <a href="/facts/Cylinder_set_measure/stLybVMq">cylinder set measure</a> on H.

<ul><li>Dudley, Richard M.; Feldman, Jacob; Le Cam, Lucien (1971). "On seminorms and probabilities, and abstract Wiener spaces". Annals of Mathematics. Second Series. 93 (2): 390–408. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970780">10.2307/1970780</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970780">1970780</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0279272">0279272</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969. <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795" target="_blank">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Structure theorem for Gaussian measures open-in-new

Structure theorem for Gaussian measures