Let $G$ be a locally compact group and $\mu$ a probability measure
on $G$. Given a unitary representation $\pi$ of $G$, let $P_\mu$ denote the
$\mu$-average $\int_G\pi(g)\,\mu(dg)$. $\mu$ is called neat if for
every unitary representation $\pi$ and every $a$ in the support of $\mu$,
$\slim_{n\to\infty}\bigl(P_\mu^n -\pi(a)^n E_\mu\bigr) =0$, where $E_\mu$ is
a canonically defined orthogonal projection. $G\/$ is called neat if every
almost aperiodic probability measure on $G$ is neat. Previously known results
show that every almost aperiodic spread out probability measure is neat, in
particular, every discrete group is neat; furthermore, identity excluding
groups, in particular, compact groups and nilpotent groups, are neat. In this
work neatness of solvable Lie groups, connected algebraic groups, Euclidian
motion groups, [SIN] groups, and extensions of abelian groups by discrete
groups is established. Neatness of ergodic probability measures on any
locally compact group is also proven. The key to these results is the
result that when $\{X_n\}_{n=1}^\infty$ is the left random walk of law
$\mu$ on $G$ and $\pi$ a unitary representation in a separable Hilbert space,
then for every $k=0,1,\dots$\,, the sequence $\pi(X_n)^{-1}P_\mu^{n-k}$
converges almost surely in the strong operator topology.