Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a Borel set of positive measure in $V$ and let $B^{V[r]}$ the Borel set in $\mathcal C^{V[r]}\times \mathcal C^{V[r]}$ with the same Borel code as $B$.

Is the section $B^{V[r]}_r=\{y\in\mathcal C^{V[r]}:(r,y)\in B^{V[r]}\}$ non-null?

I know that the set $C=\{x\in\mathcal C^V : m(B_x)>0\}$ is non-null, by Fubini's theorem, and also, by randomness, $r\in C^{V[r]}$. The answer is yes if $C^{V[r]}=\{x\in\mathcal C^{V[r]}: m(B^{V[r]}_x)>0\}$, but I do not know how this could be justified.


1 Answer 1


No, let $(r,y)\in B $ iff $r (0)=0$. Then if $r $ is a random real with $r (0)=1$ you have a counterexample.

  • $\begingroup$ Thank you. I'm afraid I have failed to isolate my real problem. I will try to reformulate it in a new question. $\endgroup$
    – Carlos
    Commented Apr 4, 2016 at 18:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.