Large cardinals and projective sets

Archive for Mathematical Logic 36 (2):137-155 (1997)
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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



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