As an application, we prove a Menchoff-Rademacher type theorem on almost everywhere convergence of series.

Although several properties and applications of orthogonal series with respect to a vector measure are known ([13, 14, 21]), the question of the almost everywhere convergence of series defined by such functions has not been studied yet.

In this section we study the almost everywhere convergence of functional series defined by (real valued) functions that are weak m-orthogonal for a vector measure m.

a,L] are enough to apply an standard almost everywhere convergence criterion (see for instance [26, III.

Note that the almost everywhere convergence with respect to a measure defined by a positive element x' do not provide m-almost everywhere convergence, since such measures are not in general Rybakov measures.

2 show that the problem of the almost everywhere convergence of weak m-orthogonal series is closely related to the calculus of estimates of 2-summing norms for the operators [[sigma].

In this paper, we are interested in the almost everywhere convergence as R [right arrow] [infinity], of the partial sums [S.

2, and since almost everywhere convergence holds for functions on S(R), which is a dense subset of [L.

Prestini, Almost everywhere convergence of the spherical partial sums for radial functions, Mh.