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We consider positive, integral-preserving linear operators acting on $L^1$space, known as stochastic operators or Markov operators. We show that, onfinite-dimensional spaces, any stochastic operator can be approximated by asequence of stochastic integral operators (such operators arise naturally whenconsidering matrix majorization in $L^1$). We collect a number of results forvector-valued functions on $L^1$, simplifying some proofs found in theliterature. In particular, matrix majorization and multivariate majorizationare related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equivalent toconvex function inequalities.

A simplified and unified generalization of some majorization results