On continuous restrictions of measurable multilinear mappings. (English. Russian original) Zbl 1344.46035
Math. Notes 98, No. 6, 977-981 (2015); translation from Mat. Zametki 98, No. 6, 930-936 (2015).
Summary: This article deals with measurable multilinear mappings on Fréchet spaces and analogs of two properties which are equivalent for a measurable (with respect to Gaussian measure) linear functional: (i) there exists a sequence of continuous linear functions converging to the functional almost everywhere; (ii) there exists a compactly embedded Banach space \(X\) of full measure such that the functional is continuous on it. We show that these properties for multilinear functions defined on a power of the space \(X\) are not equivalent; but property (ii) is equivalent to the apparently stronger condition that the compactly embedded subspace is a power of the subspace embedded in \(X\).
MSC:
| 46G12 | Measures and integration on abstract linear spaces |
| 46G25 | (Spaces of) multilinear mappings, polynomials |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
Keywords:
measurable multilinear form; measurable bilinear form; Gaussian measure; compact embedding; Banach space; Radon probability measureReferences:
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