The restriction of an irreducible unitary representation π of a real reductive group G to a reductive subgroup H decomposes into a direct integral of irreducible unitary representations τ of H with multiplicities m(π,τ) ∈ N ∪ {∞}. We show that on the smooth vectors of π, the direct integral is pointwise defined. This implies that m(π,τ) is bounded above by the dimension of the space Hom _H(π ^∞| _H,τ ^∞) of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between these two concepts of multiplicity.