Exercise 14, § 2

Let X be a locally compact space, $\mu$ a bounded positive measure on X such that $\mu(X) = 1$, and $f: X\rightarrow X$ a $\mu$-measurable mapping such that $f(\mu) =\mu$. Denote by $u_f$ the unitary endomorphism of $L^2_{\mathbf{C}}(X, \mu)$ defined by $\varphi \mapsto f\circ \varphi ($cf. Exercise 16 of I, p. 190). For every $n\in \mathbf{N}$, write $f^n=f^{\circ n}$ (mapping composed n times), and for every subset A of X, write $f^{-n}(A) =\overset{-1}{g}(A)$ with $g=f^n$.

One says that $f$ is weakly mixing relative to $\mu$ if, for all $\mu$-measurable subsets A and B of X, one has

1 $^N-_n$

lim $\sum|\mu(A\cap f(B)-\mu(A)\mu(B)|= 0$.

$n=0$

a) Prove that the following conditions are equivalent:

(i) The mapping $f$ is weakly mixing relative to $\mu$;

(ii) For all $\varphi_1$ and $\varphi_2\in L^2_{\mathbf{C}}(X, \mu)$, one has

1 $^Nn$

lim $\sum|\langle \varphi_1|u_f(\varphi_2)\rangle - \langle 1|\varphi_2\rangle \langle \varphi_1|1\rangle |= 0$;

$n=0$

(iii) The mapping $f\times f: X\times X\rightarrow X\times X$ is $(\mu\otimes \mu$)-ergodic (cf. Exercise 16 of I, p. 190) ;

(iv) The eigenspace corresponding to the eigenvalue 1 is of dimension 1 and, for every $\lambda \in \mathbf{C}-\{1\}$, the eigenspace of $u_f$ corresponding to $\lambda$ is reduced to 0.

(To prove that (iii) implies (i), use Exercise 64 of II, p. 300.)

b) Consider the example of question e) of Exercise 16 of I, p. 190. Prove that the mapping $f$ is weakly mixing relative to $\mu$.

c) Consider the example of question f ) of Exercise 16 of I, p. 190. Prove that the mapping $f$ is not weakly mixing relative to $\mu$.