About the smallest sigma field under certain conditions.

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $A, B \in \mathcal {F}$ with $A \cap B = \emptyset$. Let $\mathcal{A} \subset \mathcal{F}$ the smallest $\sigma$-field containing $A$ and not containing $B$ and $\mathcal {B} \subset \mathcal{F}$ the smallest $\sigma$-field containing $B$ and not containing $A$.

Is it true that $\mathcal{F}$ is the smallest sigma algebra containing $\mathcal {A} \cup \mathcal {B}$? And if $|\Omega|=\infty$ ?

There is a counter example?

Thank´s.

-
It is possible that neither $\mathcal{A}$ nor $\mathcal{B}$ exist; e.g., if $A=\Omega-B$. – Arturo Magidin Feb 6 '12 at 19:32
I'm not sure I understand: if $A\neq \Omega\setminus B$, then $\mathcal A=\{\emptyset, A,\Omega\setminus A,\Omega\}$, $\mathcal B=\{\emptyset, B,\Omega\setminus B,\Omega\}$, and the smallest $\sigma$-field containing $\mathcal A\cup\mathcal B$ is finite. So taking $\mathcal F$ infinite we get a counter-example. – Davide Giraudo Feb 6 '12 at 19:33
@DavideGiraudo: I would make that an answer... – Arturo Magidin Feb 6 '12 at 19:40

I'm not sure I understand: if $A\neq \Omega\setminus B$, then $\mathcal A=\{\emptyset,A,\Omega\setminus A,\Omega\}$, $\mathcal B=\{\emptyset,B,\Omega\setminus B,\Omega\}$, and the smallest $\sigma$-field containing $A\cup B$ is necessary finite. So taking $\mathcal F$ infinite we get a counter-example.

It's not even true when $\Omega$ is finite: take $\Omega:=\{a,b,c,d\}$, $\mathcal F=\mathcal P(\Omega)$, $A=\{a\}$, $B=\{b\}$. Then the smallest $\sigma$-algebra containing $\mathcal A\cup\mathcal B$ is $\{\emptyset,\{a\},\{b\},\{b,c,d\},\{a,c,d\},\{c,d\},\{a,b\},\Omega\}\neq \mathcal F$ since $\{c\}$ is not in the previous $\sigma$-field.

-
+1. One can even get counterexamples with $\Omega$ of size $3$ and $A$ and $B$ nonempty, and with $\Omega$ of size $2$ if empty sets are allowed... – Did Feb 6 '12 at 20:30