The following appears in On definability of non measurable sets, Harvey Friedman, Canadian Journal of Mathematics, Vol. 32, No. 3, 1980.

Let $M$ be the Solovay's model for $ZF + DC + V = L(R) +$ every set of reals is Lebesgue measurable etc. Let $\kappa$ be a regular cardinal of cofinality bigger than $\omega_1$ in $M$. Then forcing with countable partial functions from $\kappa$ to $2$ gives a model $N$ which satisfies choice and the statement: "Every definable, with ordinal and real parameters, set of sets of reals of size less than continuum has only Lebesgue measurable sets".