It is an open problem to determine for which maps $f$, anycompact invariant set $K$ carries an ergodic invariant measure of the sameHausdorff dimension as $K$. If $f$ is conformal and expanding, then itis a known consequence of the thermodynamic formalism thatsuch measures do exist.(We give a proof here under minimal smoothness assumptions.)If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying$\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class ofinvariant sets $K$, we show that ergodic invariant measures of dimensionarbitrarily close to the dimension of $K$ do exist. The proof isbased on approximating $K$ by self-affine sets.