  This article is devoted to a study of majorization based on semi-doubly
stochastic operators (denoted by $S\mathcal{D}(L^1)$) on $L^1(X)$ when $X$ is a
$\sigma$-finite measure space. We answered Mirsky's question and characterized
the majorization by means of semi-doubly stochastic maps on $L^1(X)$. We
collect some results of semi-doubly stochastic operators such as a strong
relation of semi-doubly stochastic operators and integral stochastic operators,
and relatively weakly compactness of $S_f=\{Sf: ~S\in S\mathcal{D}(L^1)\}$ when
$f$ is a fixed element in $L^1(X)$ by proving equi-integrability of $S_f$.
