  Given sufficiently many components, it is often cited that finite mixture
models can approximate any other probability density function (pdf) to an
arbitrary degree of accuracy. Unfortunately, the nature of this approximation
result is often left unclear. We prove that finite mixture models constructed
from pdfs in $\mathcal{C}_{0}$ can be used to conduct approximation of various
classes of approximands in a number of different modes. That is, we prove
approximands in $\mathcal{C}_{0}$ can be uniformly approximated, approximands
in $\mathcal{C}_{b}$ can be uniformly approximated on compact sets, and
approximands in $\mathcal{L}_{p}$ can be approximated with respect to the
$\mathcal{L}_{p}$, for $p\in\left[1,\infty\right)$. Furthermore, we also prove
that measurable functions can be approximated, almost everywhere.
