Consider two compact metric spaces $X$ and $Y$, and two Borel probability measures $\mu$ and $\nu$ on $X$ and $Y$, respectively. Suppose we have two ergodic $\mathbb{Z}$ actions $T\curvearrowright (X,\mu)$ and $S\curvearrowright (Y,\nu)$. We can define a $\mathbb{Z}^2$-action on $(X\times Y,\mu\times\nu)$ by $(n,m)\mapsto T^n\times S^m$.
How would I got about showing that this action is also ergodic? Since $\mathbb{Z}^2$ is locally compact and $X\times Y$ is compact, it suffices to show that
$$\forall n\forall m\left((T^n \times S^m)E=E\right)\implies (\mu\otimes \nu)(E)\in\{0,1\}$$
This is easy to see for rectangles $A\times B$, and since the above condition is clearly closed by complements, it would be enough to show that if $\{A_n\}_{n=1}^\infty$ satisfy the above condition, then so does $\bigcup_n A_n$, but I'm not sure how to do that, or even if I'm on the right track. A hint would be very helpful.
There's an easier way to show this, let $f(x,y)$ be an invariant function, approximate $f$ by a linear combination of functions define either on $x$ or on $y$, $$f(x,y)=\sum_{n=0}^\infty h_n(x)g_n(y)$$ Fix $y$, since $f(x,y)$ is invariant under the action of $T^n\times Id$, by the ergodicity of $X$ the function $f(\cdot,y)$ is constant. Now fix $x$ and consider the action of $Id\times S^m$ we have that $f(x,\cdot)$ is constant, and therefore $f$ is constant.