On Continuous Restrictions of Measurable Multilinear Mappings

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 $X$.