  In this paper, we characterize compatibility of distributions and probability
measures on a measurable space. For a set of indices $\mathcal J$, we say that
the tuples of probability measures $(Q_i)_{i\in \mathcal J} $ and distributions
$(F_i)_{i\in \mathcal J} $ are {compatible} if there exists a random variable
having distribution $F_i$ under $Q_i$ for each $i\in \mathcal J$. We first
establish an equivalent condition using conditional expectations for general
(possibly uncountable) $\mathcal J$. For a finite $n$, it turns out that
compatibility of $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ depends on the
heterogeneity among $Q_1,\dots,Q_n$ compared with that among $F_1,\dots,F_n$.
We show that, under an assumption that the measurable space is rich enough,
$(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ are compatible if and only if
$(Q_1,\dots,Q_n)$ dominates $(F_1,\dots,F_n)$ in a notion of heterogeneity
order, defined via multivariate convex order between the Radon-Nikodym
derivatives of $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ with respect to some
reference measures.
  We then proceed to generalize our results to stochastic processes, and
conclude the paper with an application to portfolio selection problems under
multiple constraints.
