TITLE: Definable non measurable set after adding one Cohen real to L
QUESTION [12 upvotes]: Suppose $x$ is a Cohen real over the constructible universe $L$. Is there a (parameter free) definable non measurable set of reals in $L[x]$?
This appeared in Will Brian's answer here where he noted that the analogous question after adding a random real has a postive answer.

REPLY [10 votes]: The paper "$\Delta^1_2$-sets of reals" by Ihoda (=Judah) and Shelah  [Annals of Pure and Applied Logic 42 (1989) 207-223] should have the result you want.
Theorem 3.1(i) tells you that every $\Delta^1_2(r_0)$ ($\Delta^1_2$ using only $r_0$ as a parameter) set is Lebesgue measurable if and only if there is a real $r$ that is random over $L[r_0]$.
In particular, all $\Delta^1_2$ (parameter free) sets of reals are measurable if and only if there is a random real over $L$.  This latter fails if we add a single Cohen real to $L$, so there is a non-measurable $\Delta^1_2$ (without parameters) set in $L[c]$ if $c$ is Cohen over $L$.
The paper addresses regularity properties of $\Delta^1_2$ sets in many different settings, and not just Lebesgue measurability.