Trivial intersection of sigma-fields and Gibbs sampling

Let $(Omega,mathcalF,P)$ be a probability space and $mathcalN$ the class of those $FinmathcalF$ satisfying $P(F)in,1$. For each $mathcalGsubsetmathcalF$, define $overlinemathcalG=sigmaigl(mathcalGcupmathcalNigr)$. Necessary and sufficient conditions for $overlinemathcalAcapoverlinemathcalB=overlinemathcalAcapmathcalB$, where $mathcalA,mathcalBsubsetmathcalF$ 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 $(Omega,mathcalF)=(mathcalX imesmathcalY,mathcalUotimesmathcalV)$ where $(mathcalX,mathcalU)$ and $(mathcalY,mathcalV)$ are measurable spaces. Let $(X_n,Y_n)_ngeq 0$ be the Gibbs-chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if $overlinesigma(X)capoverlinesigma(Y)=mathcalN$, or equivalently if and only if $P(Xin U)P(Yin V)=0$ whenever $UinmathcalU$, $VinmathcalV$ and $P(U imes V)=P(U^c imes V^c)=0$. The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset $S_0subsetOmega$, in case $mathcalF=mathcalUotimesmathcalV$ is countably generated and $P$ absolutely continuous with respect to a product measure.