Set of independent sets is a sigma-algebra? I was wondering whether the set $\{E \in \mathscr{E};A,B,E \text{ are independent} \}$ is always a sigma-algebra, when $A,B$ are two independent sets in some probability space? I know that it is a Dynkin-system but is it also true that it is a sigma-algebra? 
Somehow I guess the answer is no, but could not find a counterexample so far.
 A: Define a probability space $\langle \Omega,\mathscr{F},P\rangle$ where 
\begin{eqnarray}
\Omega&=&\mathbf{2}^{3} \\
\mathscr{F}&=&\mathcal{P}\Omega \\
\mu(A)&=&\frac{\vert A\vert}{8}.
\end{eqnarray}
Now, let $A$ and $B$ be defined by 
\begin{eqnarray}
A&=&\pi_1^{-1}\{0\}=\big\{\langle 0,x,y\rangle\in\Omega\big\vert x,y\in\{0,1\}\big\}\\
B&=&\pi_2^{-1}\{0\}=\big\{\langle x,0,y\rangle\in\Omega\big\vert x,y\in\{0,1\}\big\}
\end{eqnarray}
so that $A$ and $B$ are independent with $P(A)=P(B)=\frac{1}{2}$, and let $\mathscr{I}$ denote the set of elements of $\mathscr{F}$ which are independent of $A$ and $B$. 
Finally, let $E_1$ and $E_2$ be:
\begin{eqnarray}
E_1 &= &\big\{\langle 0,0,0\rangle,\langle 1,1,0\rangle,\langle 1,0,0\rangle,\langle 0,1,0\rangle\big\} \\
E_{2}&=&\big\{\langle 0,0,0\rangle,\langle 1,1,1\rangle,\langle 1,0,1\rangle,\langle 0,1,1\rangle\big\}.
\end{eqnarray}
Then $E_1$ and $E_2$ are both independent of $A$ and $B$ (and $A\cap B$) and are thus elements of $\mathscr{I}$.  But $$E_1\cap E_2=\big\{\langle 0,0,0\rangle\big\}\subseteq A\cap B$$ and so $E_1\cap E_2$cannot be independent of either $A$ or $B$. 
It follows that $\mathscr{I}$ is not a $\pi$-system, and hence not a $\sigma$-algebra.
