Comment by Elliot Glazer on Why is inner model theory evidence for consistency of large cardinals?

@n901 All reasonable formulations of "all sets of reals are Lebesgue measurable" are equivalent in ZF, and prove the $\sigma$-additivity of $\lambda$ (see mathoverflow.net/a/393162/109573). We get a relative consistency proof over ZF by building the Solovay model relative to the Levy collapse of $\omega_{\omega}$ and adapting Solovay's argument to show $\lambda^*$ and $\lambda_*$ agree on subsets of the unit interval.