Where does this proof of ergodicity fail?

Question

I am a bit confused and was hoping the folk of MSE would help (and point out where the error is made).

Suppose $(X,\mathcal{B},\mu,R)$ and $(Y,\mathcal{C},\nu,S)$ are ergodic measure preserving dynamical systems. Let $m=\mu\otimes \nu$ and define $T:X\times Y\to X\times Y$ by $T(x,y)=(Rx,Sy)$. Note that if $A\times B\subset X\times Y$ is measurable with $T^{-1}(A\times B)=A\times B$, then $R^{-1}A=A$ and $S^{-1}B=B$ and so by ergodicity, $\mu(A),\nu(B)\in\{0,1\}$. It follows that $m(A\times B)\in\{0,1\}$ and so $m$ is ergodic.

On the other hand, this is clearly false - take $T(x,y)=x-y$ on the torus $\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$. This function is not constant, but it is invariant: it is easy to check that $T(x+\alpha, y+\alpha)=T(x,y)$ for any $\alpha$.

Thanks in advance!

score 1 · Accepted Answer · answered Jul 09 '18 at 16:18

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Not every measurable subset of $X\times Y$ has the form $A\times B$! Indeed, in your example, for instance, the set of $(x,y)$ such that $|x-y|<1/4$ is measurable and is a counterexample to ergodicity which is not of that form.

answered Jul 09 '18 at 16:18

Eric Wofsey

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But is it not the case that sets of the form $A\times B$ form a generating $\pi$-system? – Mr Martingale Jul 09 '18 at 23:39
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That is true. So what? – Eric Wofsey Jul 10 '18 at 00:16
Is this not sufficient to deduce measurability w.r.t the $\sigma$-algebra? – Mr Martingale Jul 10 '18 at 11:35
IN ADDITION TO ABOVE COMMENT: Does this not naturally extend to ergodicity? I.e. any $T$ invariant set in the product $\sigma$-algebra could be written as intersections/unions/etc of sets of the form $A_i\times B_i$? Or am I horribly mistaken (which I probably am) – Mr Martingale Jul 10 '18 at 11:41
Any measurable set can be written as a combination of sets of the form $A_i\times B_i$, but that does not mean the individual sets $A_i\times B_i$ used must be invariant in order for the set to be invariant. – Eric Wofsey Jul 10 '18 at 16:03
Ah okay, that makes sense. Thanks! – Mr Martingale Jul 10 '18 at 16:07

Where does this proof of ergodicity fail?

1 Answers1