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It is shown that a Gaussian measure in a given infinite-dimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator. This covers a similar statement made earlier in [Lee and Wasilkowski, Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2(1) (1986) 12-43.] for the case of finite-rank operators.

For a topological abelian group X we topologize the group c 0 .X / of all X-valued null sequences in a way such that when X D R the topology of c 0 .R/ coincides with the usual Banach space topology of the classical Banach space c 0 . If X is a nontrivial compact connected metrizable group, we prove that c 0 .X / is a non-compact Polish locally quasi-convex group with countable dual group c 0 .X /^. Surprisingly, for a compact metrizable X, countability of c 0 .X/^leads to connectedness of X.Our principal application of the above results is to the class of locally quasi-convex Mackey groups (LQC-Mackey groups). A topological group .G; / from a class G of topological abelian groups will be called a Mackey group in G or a G -Mackey group if it has the following property: if is a group topology in G such that .G; / 2 G and .G; / has the same character group as .G; /, then Ä .Based upon the results obtained for c 0 .X/, we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC-Mackey. Namely, we show that for a connected compact metrizable group X ¤ ¹0º, the group c 0 .X /, endowed with the topology induced from the product topology on X N , is a metrizable precompact connected group which is not a Mackey group in LQC.Since metrizable locally convex spaces always carry the Mackey topology -a wellknown fact from Functional Analysis -, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.

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