Absolutely continuous representing measures for R(X)

Let X be a compact nowhere dense subset of the complex plane C, and let R(X) be the uniform closure of the rational functions having no poles on X. We present a short proof of the fact that each non-peak point for R(X) admits a representing measure absolutely continuous with respect area. If X happens to have finite perimeter, then every non-peak point admits a representing measure absolutely continuous with respect to arc length, provided the corresponding integral is taken in a principal value sense.