Title says it all, so I'll just repeat it: Does every Lebesgue measurable set have the Baire property?
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No. (assuming choice, so that there are some sets without the Baire property).
Some hints: Any set with the property of Baire which is not meager, has a subset that fails the property of Baire. Next, find a Lebesgue null set, with the property of Baire, but not meager.
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.