The Cantor-Lebesgue and Denjoy-Luzin properties for double systems of functions (Q1261257)

The double series (1) \(\sum^ \infty_{m=1}\sum^ \infty_{n=1} a_{mn}\varphi_{mn}(x)\) is considered, where \(\{a_{mn}\}\) is a double sequence of real numbers \(\Phi=\{\varphi_{mn}(x); m,n=1,2,\dots\}\) is a double system of measurable functions defined on a finite positive measure space \((X,{\mathcal F},\mu)\). It is assumed that the series (1) convergences or is Cesàro summable on a set of positive measure or almost everywhere (in Pringsheim's or regular sense). Hence properties of \(\{a_{mn}\}\) \((m,n=1,2,\dots)\) are concluded.