TITLE: Does every Lebesgue measurable set have the Baire property?
QUESTION [11 upvotes]: Title says it all, so I'll just repeat it:
Does every Lebesgue measurable set have the Baire property?

REPLY [4 votes]: Just to make the solution from the comments to GEdgar's answer more explicit: The set of real numbers can be partitioned as $\mathbb R = M \cup N$, where $M$ is meager and $N$ is Lebesgue null.  ($N$ is the dense $G_\delta$ described by t.b. above.) 
Now every subset of $N$ is Lebesgue measurable, but (using AC) one can find a subset without the Baire property.