  We study monotone homomorphisms on the semigroup of probability measures on
$\mathbb{R}$, by which we mean maps to the reals that are monotone with respect
to the stochastic order and additive under convolution. We show that scalar
multiples of the expectation are the unique monotone homomorphisms on the
semigroup of measures with finite $p$-th moment, for any $1 \le p < \infty$. We
also prove that the entire semigroup of probability measures admits no non-zero
monotone homomorphism.
