What is the Sigma Algebra generated by Jordan measurable sets?

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 $$J$$?

• This jstor.org/stable/44153840 seems to contain the answer, if you have access to it. I don't. – daw Oct 31 '18 at 14:56
• @daw: If anyone without access is interested, I have a copy of the original journal volume (from personal subscription) from which I can make photocopy, followed by a .pdf of that photocopy. Send me an email request. My email can be deduced from information in my mathematical stack exchange profile. – Dave L. Renfro Oct 31 '18 at 16:53
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.