This talk is about work in progress with Juan Aguilera (TU Wien) and Keita Yokoyama (Tohoku University) on the foundations of mathematics, studying the measurability of analytical sets, with the method of random forcing.
Yu showed that assuming ATR0 was sufficient (and necessary) to prove that the measure of any coded Borel set exists. We answer Simpson's question about the correct subsystem to prove analytic measurability (originally demonstrated by Lusin).
By drawing inspiration from Solovay's construction of a model of ZF where every set is Lebesgue measurable, we use ATR0 and, in particular, the method of pseudo-hierarchies to construct a non-standard model providing us with all the necessary transfinite information about a given analytical set A.
We then show that the scheme of induction for analytical formulae suffices to prove the regularity of A together with the implementation of our technique. If one assumes the scheme of analytical comprehension in turn, we get the classical result of measurability for A.
We also show that the respective reversals hold.