Trivial intersection of sigma-fields and Gibbs sampling

Let $(\Omega,\mathcal{F},P)$ be a probability space and
$\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying
$P(F)\in\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define
$\overline{\mathcal{G}}=\sigma\bigl(\mathcal{G}\cup\mathcal{N}\bigr)$.
Necessary and sufficient conditions for
$\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline{\mathcal{A}\cap\mathcal{B}}$,
where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are
sub-$\sigma$-fields, are given. These conditions are then applied to
the (two component) Gibbs sampler. Suppose $X$ and $Y$ are the
coordinate projections on
$(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal{U}\otimes\mathcal{V})$
where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$
are measurable spaces. Let $(X_n,Y_n)_{n\geq 0}$ be the Gibbs-chain
for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if
$\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or
equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever
$U\in\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times
V^c)=0$. The latter condition is also equivalent to ergodicity of
$(X_n,Y_n)$, on a certain subset $S_0\subset\Omega$, in case
$\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated
and $P$ absolutely continuous with respect to a product measure.