We assume that for every real $x$, $L[x]$ only contains countably many reals.

Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that

1. For any reals $x\in I$ and $y$, $y\in L[x]$ implies $y\in I$; and
2. For any finite $F\subseteq X$, there is a real $z\in I$ so that $F\subseteq L[z]$.

The question is

Question: Given a null set $X$ only containing $L$-random reals, must the $L$-ideal $I$ generated by $X$ be null?

Note that, given the set $X$ as in the question, the $L$-upward closure $U_X=\{y\mid \exists x\in X(x\in L[y])\}$ must be null.