Analytic sets are projections of Borel sets, and are known to be Lebesgue measurable (in fact universally measurable). The question of whether measurability of analytic sets can be shown in some standard subsystem of second order arithmetic has been discussed on MO, and apparently was recently solved: analytic measurability is equivalent over $\mathrm{ATR_0}$ to $\Pi^1_1$-CA$_0$. (I am an outsider and not sure why $\mathrm{ATR_0}$ rather than $\mathrm{RCA_0}$ is chosen to be the base theory. Is it the most convenient place to develop measure theory?)
The next natural question is measurability of $\mathbf{\Sigma^1_2}$ sets, namely projections of complements of analytic sets. $\mathbf{\Sigma^1_2}$-measurability is independent of $\mathsf{ZFC}$, but does not have consistency strength over $\mathsf{ZFC}$ (in contrast to $\mathbf{\Sigma^1_3}$-measurability, which needs an inaccessible by Shelah). The consistency proof I know is via showing $\mathsf{MA}_{\aleph_1}$ implies the union of $\aleph_1$ many null sets is null, and that in $\mathsf{ZFC}$ any $\mathbf{\Sigma^1_2}$ set is the union of $\aleph_1$ many Borel sets. To get $\mathsf{MA}_{\aleph_1}$ we typically perform an iterated forcing of length $\aleph_2$, and I don't know if something like that can be carried out in second order arithmetic.
Questions: Does $\mathbf{\Sigma^1_2}$-measurability have consistency strength over $\Pi^1_1$-CA$_0$? Over the full second order arithmetic?
