  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
$\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. 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
$(\mathcal{Y},\mathcal{V})$ are measurable spaces. 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
$S_0\subset\Omega$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is
countably generated and $P$ absolutely continuous with respect to a product
measure.
