Does every Lebesgue measurable set have the Baire property?

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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.