# Error in proof of $L_p$ equivalent weak-mixing conditions.

In page 45 of Peter Walters' "Introduction to Ergodic theory" there is a theorem which states that:

Theorem: $$T:X \to X$$ is weak-mixing if and only if for all $$f,g \in L_2(X)$$ we have that $$\lim_{n\to \infty} \frac{1}{n} \sum^{n-1}_{i=0} \Big| \int U_T^n(f) \cdot g \: dm - \int f \: dm \int g \: dm \Big| = 0$$ where $$U_T(f)$$ is defined by $$U_T (f)(x) = (f\circ T)(x)$$

The only if part of this proof follows trivially (apparently) by taking $$f = \chi_A, \: g = \chi_B$$ such that \begin{align} 0 & = \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \Big| \int|\chi_A \circ T^n \cdot \chi_B| dm - \int \chi_A dm \int \chi_B dm \Big|\\ & = \lim_{n \to \infty} \frac{1}{n} \sum^{n-1}_{i=0} \Big|m( T^nA \cap B) - m(A)m(B) \Big| \end{align}

However this final line does not actually imply weak mixing as far as I can tell since the weak-mixing condition is

$$\lim_{n \to \infty} \frac{1}{n} \sum^{n-1}_{i=0} \Big|m( T^{-n}A \cap B) - m(A)m(B) \Big|.$$

That is, we need the statement to hold for $$T^{-n}$$ not $$T^n$$. If anyone could help me see why the difference doesn't matter, or has an alternative proof it would be very helpful.

• You just made a mistake in writing $\chi_A\circ T^{n}$ as $\chi_{T^{n}(A)}$. – Kavi Rama Murthy Apr 28 '19 at 5:19
• Yep I see. Thanks a lot. – Debreu Apr 28 '19 at 8:58

I'll answer my own question. The expression with $$T^n$$ is wrong. Notice $$U_T (\chi_B) = \chi_{T^{-1}B}$$