We show that the class of sets having polynomial size circuits, P/poly, has EXP/sup NP/-measure zero under each of the following two assumptions: EXP/sup NP//spl ne/ZPP(/spl Sigma//sub 2//sup p/)(which holds if the polynomial time hierarchy does not collapse to ZPP(/spl Sigma//sub 2//sup p/)), or NP is not small (does not have EXP-measure zero).

We consider the resource bounded measure of polynomial-time learnable subclasses of polynomial-size circuits. We show that if EXP/spl ne/MA, then every PAC-learnable subclass of P/poly has EXP-measure zero. We introduce a nonuniformly computable variant of resource bounded measure and show that, for every fixed polynomial q, any polynomial-time learnable subclass of circuits of size q has measure zero with respect to P/poly. We relate our results to the question whether Boolean Circuits are polynomial-time learnable.