Local entropy averages and projections of fractal measures

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X, Y [0, 1] are closed and invariant, respectively, under m mod 1 and n mod 1, where m, n are not powers of the same integer, then, for any t = 0,