Generic sections of non-null sets are non-null

Question

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.

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.