Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null $F_\sigma$ set?

The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be written as a union of a Borel set and a subset of a measure $0$ Borel set. But my question is, what is an example of a Lebesgue measurable set which cannot be written as a union of a Borel set and a subset of a measure $0$ $F_\sigma$ set?

Or does no such example exist?

Solution 1:

I found an example in this journal paper. Let $\beta$ be a Bernstein set, i.e. a subset of $\mathbb{R}$ such that both it and its complement intersects every uncountable closed subset of $\mathbb{R}$. (This post describes how to construct such a set using the axiom of choice.) And let $\gamma$ be a dense measure-$0$ $G_\delta$ subset of the fat Cantor set. (This answer describes how to construct such a set.)

Then $\beta\cap\gamma$ is a Lebesgue measurable set which cannot be written as a union of a Borel set and a subset of a measure $0$ $F_\sigma$ set. This is proven in the linked paper.