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