A real $x\in2^\omega$ is random (or Solovay-random) over a model $M$ iff it's obtained via forcing by closed sets of reals, of positive measure, coded in $M$. Similarly, a pair $(x,y)$ of reals is random over a model $M$ iff it's obtained via forcing by closed sets in $2^\omega\times2^\omega$, of positive product measure, still coded in $M$. Thus such a pair is **not** a product-forcing pair. (To establish this, one has to define a closed set $X\subseteq2^\omega\times2^\omega$, of positive product measure, which does not include any positive-measure rectangle, not an easy exercise!) Rather $(x,y)$ is a random pair over a model $M$ iff $x$ is random over $M$ and $y$ is random over $M[x]$ iff *vice versa*.


**Question**. In spite of not being product-forcing-generic, prove that if $(x,y)$ is a random pair over a model $M$ then $M[x]\cap M[y]=M$. 


A proof of $M[x]\cap M[y]\cap2^\omega\subseteq M$ can be manufactured by reduction, via continuous reading of names, to the claim that if $X\subseteq2^\omega\times2^\omega$, is a closed set of positive product measure and $f,g:2^\omega\to2^\omega$ are continuous maps such that $f(x)=g(y)$ for almost all $(x,y)\in X$, then $X=X'\cup\bigcup\_nX\_n$, where $X'$ is null and each $X\_n$ is Borel non-null on which both $f$ and $g$ are constants. What about the general case?



Solovay-random pairs of reals
Suppose $\kappa$ is the least ordinal such that for some $A \in 2^{\kappa} \cap M[x] \cap M[y]$, $A \notin M$. You already have that $\kappa = cf(\kappa) \geq \omega\_1$. Let $\tau$ be a Random name for $A \in M[x]$. WLOG, assume that the empty condition forces $\tau \in 2^{\kappa} \setminus M$ and all initial segments are in $M$. In $M$, choose $\langle (p\_i, a\_i) : i < \kappa \rangle$ such that $p\_i$ forces $a\_i = \tau \upharpoonright i$. Since random forcing is $\kappa$-Knaster, there exists $X \in [\kappa]^{\kappa}$ such that $\{p\_i : i \in X\}$ has pairwise compatible conditions so that $a = \bigcup \{a\_i: i \in X\} \in 2^{\kappa}$. Using cccness, choose a condition $p$ that forces $|\{i \in X: p\_i \in G\}| = \kappa$. But now $p$ forces $\tau \in M$: Contradiction. 


