Large cardinals and projective sets

Abstract.

We investigate measure and category in the projective hierarchie in the presence of large cardinals. Assuming a measurable larger than $n$ Woodin cardinals we construct a model where every $\Delta ^1_{n+4}$-set is measurable, but some $\Delta ^1_{n+4}$-set does not have Baire property. Moreover, from the same assumption plus a precipitous ideal on $\omega _1$ we show how a model can be forced where every $\Sigma ^1_{n+4}-$set is measurable and has Baire property.