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For your second definition of Vitali set, I have a weak partial answer. Namely the existence of a Bernstein set does not imply the existence of a $T$-Vitali set. The answer can be found in logic blog …

Not sure whether the following answers your question, but they might be helpful. Fix any number $n\geq 2012$. 1 For any $e_0$ so that $\Phi_{e_0}^X=X_0$ where $X_0$ is the unique real so that $X=X_0 …

I want to take this opportunity to give an application of algorithmic randomness theory to this area. Accidentally I am working on the similarity problem recently and found some interesting applicatio …

It is well known that for any continuous function $f$, $f$ sends all measurable sets to measurable ones if and only if $f$ has Luzin's-(N)-property. I.e. $f$ sends all null sets to null ones. By [1], …

Here is a quite short example to show that you question cannot have a positive answer. Assume that $V$ is a Sacks extension of constructible universe $L$. Then the set of constructible reals $A=(\math …

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 For any reals $x\in …

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 Mar …

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known th …

There is a pure recursion theoretical proof of the result. The idea is as follows: By Spector-Gandy theorem, a lightface Borel set $(x,y)$ is an r.e set over $L_{\omega_1^{CK}}[x,y]$. If there are at …

Here is an answer to your question. Assume $V=L$, there is a null $\Pi^1_1$-set which is not in $\sigma(J)$. Proof: Let $A=\{x\mid x\in L_{\omega_1^x}\}$ be the $\Pi^1_1$-null set. Now suppos …