  The restriction of an irreducible unitary representation $\pi$ of a real
reductive group $G$ to a reductive subgroup $H$ decomposes into a direct
integral of irreducible unitary representations $\tau$ of $H$ with
multiplicities $m(\pi,\tau)\in\mathbb{N}\cup\{\infty\}$. We show that on the
smooth vectors of $\pi$, the direct integral is pointwise defined. This implies
that $m(\pi,\tau)$ is bounded above by the dimension of the space
$\operatorname{Hom}_H(\pi^\infty|_H,\tau^\infty)$ of intertwining operators
between the smooth vectors, also called symmetry breaking operators, and in
this way provides a precise relation between these two concepts of
multiplicity. This statement might have been common knowledge, but we could not
find a reference in the literature.
