A survey about the most important properties of extended three weak majorization relations $\prec_w$, $\prec_s$ and $\prec^{ws}$ determined by stochastic operators on the discrete Lebesgue space $\ell^1(I)$, is presented. In the second part, linear preservers of considered majorization relations, when $I$ is an infinite set, are characterized and many examples with concrete matrix forms of linear preservers are given.