  The study of uniformly distributed measures was crucial in Preiss' proof of
his theorem on rectifiability of measures with positive density. It is known
that the support of a uniformly distributed measure is an analytic variety. In
this paper, we provide quantitative information on the rectifiability of this
variety. Tolsa had already shown that $n$-uniform measures have Big Pieces of
Lipschitz Graphs(BPLG) . Here, we prove that a uniformly distributed measure
has BPLG locally.
