  We note that the well-known result of Von Neumann \cite{von} is not valid for
all doubly substochastic operators on discrete Lebesgue spaces $\ell^p(I)$,
$p\in[1,\infty)$. This fact lead us to distinguish two classes of these
operators. Precisely, the class of proper doubly substochastic operators on
$\ell^p(I)$ is isolated with the property that an analogue of the Von Neumann
result on operators in this class is true. The submajorization relation
$\prec_s$ on the positive cone $\ell^p(I)^+$, when $p\in[1,\infty)$, is
introduced by proper substochastic operator and it is provided that
submajorization may be considered as a partial order. Two different shapes of
linear preservers of submajorization $\prec_s$ on $\ell^1(I)^+$ and on
$\ell^p(I)^+$, when $I$ is an infinite set, are presented.
