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This article is devoted to a study of majorization based on semi-doublystochastic 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 characterizedthe majorization by means of semi-doubly stochastic maps on $L^1(X)$. Wecollect some results of semi-doubly stochastic operators such as a strongrelation 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 $S_f$.

Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$