The measure of ideals generated by random reals

Question

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.

Answer 1

The question has a negative answer. The technique is essentially due to Jockusch and Posner.

Proof: Let $x$ be a real in which every constructible real is recursive. Now $$A=\{r\mid r\mbox{ is Martin-L\" of random relative to }x \wedge x\oplus r\geq_T x',\mbox{ the Turing jump of }x.\}$$ Then $A$ is null, $r\oplus x$ ranges over the upper cone $\geq_T x'$, and only contains $L$-random reals. Now for any $z\geq_T x'$ and $r \in A$, we have that

(1). $r\triangle x \in A$; and

(2). $(r\triangle x) \oplus r\equiv_T x\oplus r\geq_T x'$.

So the $L$-ideal generated by $A$ is $\mathbb{R}$.