Approximation of linear functionals on the space with convex measure

TL;DR

This paper investigates the conditions under which measurable linear functionals on topological vector spaces with convex measures are equivalent to limits of continuous linear functionals, extending known results from Gaussian measures.

Contribution

It proves the equivalence of measurable and limit-based linear functionals for certain classes of convex measures, generalizing Gaussian measure results.

Findings

Equivalence holds for specific classes of convex measures.

Extends Gaussian measure results to broader convex measure contexts.

Provides theoretical foundation for functional analysis with convex measures.

Abstract

There are two definitions of the measurable functional on the topological vector space: as a linear and measurable real-valued function and as a pointwise limit of the sequence of the continious linear functionals. In general case they are not equivalent, but in some cases it is so, for example, in the case of gaussian measures. There is one natural generalization of the gaussian measures - the convex measures. In this paper this equivalence was proved for the some classes of convex measures.