Trivial intersection of $\sigma$-fields and Gibbs sampling

Let $(\mathrm{\Omega },\mathcal{F},P)$ 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 $\bar{\mathcal{G}}=\sigma (\mathcal{G}\cup \mathcal{N})$. 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 $\mathcal{V}$)$where$($\mathcal{X}$,$\mathcal{U}$)$and$($\mathcal{Y}$,$\mathcal{V}$)$aremeasurablespaces\mathrm{.}Let$(X_n,Y_n)_{n\geq0}$ 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 $S_0\subset \mathrm{\Omega }$, in case $\mathcal{F}=\mathcal{U}\otimes \mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure.