We consider two natural definitions for the notion of a
dynamical system being mixing relative to an invariant sub
$\sigma$-algebra $\mathcal{H}$. Both concern the convergence of
¶
\[|E(f\cdot g\circ T^n|\mathcal H)-E(f|\mathcal H)E(g\circ T^n|\mathcal
H)|\to 0\]
¶ as $|n|\to \infty$ for appropriate $f$ and $g$. The weaker
condition asks for convergence in $L^1$ and the stronger for convergence
a.e. We will see that these are different conditions. Our goal is to
show that both these notions are robust. As is quite standard we show
that one need only consider $g=f$ and $E(f|\mathcal H)=0$, and in this
case $|E(f\cdot f\circ T^n|\mathcal H)|\to 0$. We will see rather
easily that for $L^1$ convergence it is enough to check an $L^2$-dense
family. Our major result will be to show the same is true for pointwise
convergence, making this a verifiable condition. As an application we
will see that if $T$ is mixing then for any ergodic $S$, $S\times T$ is
relatively mixing with respect to the first coordinate sub
$\sigma$-algebra in the pointwise sense.