Point Evaluations and Polynomial Approximation in the Mean

We construct an example of a Borel subset E of the unit disk such that area measure restricted to E, which we denote A _E , has the property that the set of bounded point evaluations for the polynomials with respect to the L ^t (A _E ) norm varies with t. We further show that E can be chosen to be a simply connected region. In the context of smooth measures, like area measure, examples of this type were unexpected.