Unlike Lebesgue measurable sets, Jordan measurable sets do not form a Sigma algebra. So my question is, what is the Sigma algebra $J$ generated by Jordan measurable sets?
All intervals are Jordan measurable, so $J$ contains all the Borel sets. But this answer shows that not all Jordan measurable sets are Borel sets, so the Borel Sigma algebra is a proper subset of $J$. And all Jordan measurable sets are Lebesgue measurable, so $J$ is a subset of the Lebesgue Sigma algebra. But are there Lebesgue measurable sets not contained in $J$?
Best Answer
I found the answer in this journal paper. The Sigma algebra generated by the Jordan measurable sets is the collection of all sets which can be written as a union of a Borel set and a subset of a measure $0$ $F_\sigma$ set. As a point of comparison, the collection of Lebesgue measurable sets is the collection of all sets which can be written as a union of a Borel set and a subset of a measure $0$ Borel set.
And the paper gives an example of a Lebesgue measurable set which is not in the Sigma algebra generated by Jordan measurable sets. 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 is not in the Sigma algebra generated by Jordan measurable sets.
But there may still be unsolved problems about this Sigma algebra, so I just posted a question on MathOverflow about it.