  In this paper we prove self-improvement properties of strong Muckenhoupt and
Reverse H\"older weights with respect to a general Radon measure on
$\mathbb{R}^n$. We derive our result via a Bellman function argument. An
important feature of our proof is that it uses only the Bellman function for
the one-dimensional problem for Lebesgue measure; with this function in hand,
we derive dimension free results for general measures and dimensions.
