  We prove for an arbitrary complex $^*$-algebra $A$ that every topologically
irreducible $^*$-representation of $A$ on a Hilbert space is finite dimensional
precisely when the Lebesgue decomposition of representable positive functionals
over $A$ is unique. In particular, the uniqueness of the Lebesgue decomposition
of positive functionals over the $L^1$-algebras of locally compact groups
provides a new characterization of Moore groups.
