# Which sets are Lebesgue measurable in $ZFC$?

The Borel sets and also the analytic sets are known to be Lebesgue Measurable in $ZFC$. The Analytic sets were defined by Suslin using the Suslin operation or the $A$-operation and Luzin proved that the Lebesgue measurable sets are closed under this operation. So the closure of the analytic sets under the Suslin operation is a class, that extends the analytic sets, in which all the sets are Lebesgue measurable.

However the sets $\Delta_2^1$ (The second level of the projective hierarchy) cannot be all be shown to be Lebesgue measurable in $ZFC$ (by Gödel's results).

Are there any known result concerning this "cap" of sets that can be proven to be Lebesgue measurable in $ZFC$?

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $$A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\},$$ and $\mathsf{ZFC}$ proves that $\phi$ and $\psi$ are equivalent. This readily follows from his arguments in his paper on the consistency of all sets of reals being measurable.

MR0265151 (42 #64). Solovay, Robert M. A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2) 92 1970 1–56.

A. Kanamori's book has details in section 14.

MR1994835 (2004f:03092). Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xxii+536 pp. ISBN: 3-540-00384-3.

As you mentioned, we cannot remove the provability requirement (since, by K. Gödel's arguments, for any real $a$, $L[a]$ is a model with a nonmeasurable $\Delta^1_2(a)$ set). On the other hand, without going beyond $\mathsf{ZFC}$ in consistency strength, larger pointclasses can be shown measurable under suitable additional axioms, such as $\mathsf{MA}$:

From $\mathsf{MA}$ (plus $\lnot\mathsf{CH}$) it follows that every $\mathbf\Sigma^1_2$ set of reals is Lebesgue measurable and has the property of Baire. This is already established in the original Martin-Solovay paper.

MR0270904 (42 #5787). Martin, D. A.; Solovay, R. M. Internal Cohen extensions. Ann. Math. Logic 2 1970 no. 2, 143–178.

For a modern exposition I suggest the Bartoszyński-Judah book.

MR1350295 (96k:03002). Bartoszyński, Tomek; Judah, Haim. Set theory. On the structure of the real line. A K Peters, Ltd., Wellesley, MA, 1995. xii+546 pp. ISBN: 1-56881-044-X.

Martin's axiom can be forced over any model of $\mathsf{ZFC}$, so it is equiconsistent with $\mathsf{ZFC}$. Even further, S. Shelah showed that $\mathsf{ZFC}$ is equiconsistent with the measurability of $\mathbf\Delta^1_3$ sets, and Judah strengthened this fact by showing that adding $\aleph_1$ random reals to a model of $\mathsf{MA}$ results in a model where all $\mathbf\Delta^1_3$ sets are Lebesgue measurable (this is described in the Bartoszyński-Judah book as well).

Nevertheless, you cannot go much further by restricting to $\mathsf{ZFC}$ consistencywise: Shelah showed that the measurability of the $\mathbf\Sigma^1_3$ sets implies the existence of inaccessible cardinals in $L$.

MR0768264 (86g:03082a). Shelah, Saharon. Can you take Solovay's inaccessible away? Israel J. Math. 48 (1984), no. 1, 1–47.

• Thanks for your answer that I am satisfied with! But now after few weeks I started to wonder about your last phrase. Could you elaborate on the fact that "larger pointclasses can be shown measurable under $MA$"? Also as Im not too familiar with $MA$ so is it really the case that it doesn't add consistency strength to $ZFC$? Commented Sep 18, 2017 at 9:54
• @Harto I have updated the answer to address your questions. Commented Sep 21, 2017 at 14:10
• Thank you! I also saw your comment earlier today but didnt have the time to read it properly. Great answer! Commented Sep 21, 2017 at 17:55