Topics from Ergodic Theory

Let $(\Omega , \mathcal{A}, P) = \left([0,1), \mathcal{B}_{[0,1)}, \lambda \right)$ where $\lambda$ is the Lebesgue measure, and let the transformation $T$ be defined by

Then show that $T$ is measurable and measure-preserving.

Let $(\Omega , \mathcal{A}) = \left([0,1), \mathcal{B}_{[0,1)}\right)$, and let the transformation $T$ be defined by

Then show that there is no probability measure $P$ on $\mathcal{B}_{[0,1)}$ such that $\mathbb {P}\left(\left\{ x\right\} \right) = 0, x \in [0,1)$, and for which the transformation $T$ is measure-preserving.

Refer to Exercise 1 and examine the transformation $T$ from the ergodicity viewpoint.

Recall (Definition 9) that a r.v. $X$ on $(\Omega , \mathcal{A}, P)$ is said to be invariant (relative to a measurable transformation $T$ on $\Omega$ into itself), if $X(\omega ) = X(T \omega ), \omega \in \Omega$. Also recall that the class $\mathcal{J}$ of ($T$-)invariant sets in $\mathcal{A}$ (i.e., the sets $A \in \mathcal{A}$ with $T^{-1} A = A$) is a $\sigma$-field (Proposition 12).

Complete the proof of the converse part of the following proposition (Proposition 16). Let $T$ be a (measurable) transformation on $(\Omega , \mathcal{A}, P)$ into itself, and let $X$ be a r.v. on the same space. Then $X$ is invariant (relative to $T$), if and only if $X$ is $\mathcal{J}$-measurable, where $\mathcal{J}$ is the $\sigma$-field of ($T$-)invariant sets in $\mathcal{A}$. The converse part asks to complete the proof of the "if" direction: that $\mathcal{J}$-measurability of $X$ implies that $X$ is invariant.

Recall the following lemma (Lemma 2). Let $T$ be a (measurable) measure-preserving transformation on $(\Omega , \mathcal{A}, P)$ into itself, and let $X$ be a r.v. on the same space for which $\mathbb {E}\left[X\right]$ exists. Then $\mathbb {E}\left[X\right](\omega ) = \mathbb {E}\left[X\right](T \omega )$.

Complete the proof of Lemma 2.

Recall (Definition 11) that a set $A \in \mathcal{A}$ is said to be invariant (relative to the process $\mathbf{X} = \left(X_{1}, X_{2}, \ldots \right)$), if there exists a set $B \in \mathcal{B}^{\infty }$ such that $A = X^{-1} B$ and such that $A = \left(X_{n}, X_{n+1}, \ldots \right)^{-1} B$ for all $n \geq 1$.

Work out the details of the proof of the following proposition (Proposition 18). The class $\mathcal{J}$ of invariant sets in $\mathcal{A}$ is a $\sigma$-field.

Recall (Definition 13) that $Z_{2}$ is said to be a function of $Z_{1}$ if for all $\omega , \omega^{\prime } \in \Omega$ with $\omega \neq \omega^{\prime }$ for which $Z_{1}(\omega ) = Z_{1}\left(\omega^{\prime }\right)$, it follows that $Z_{2}(\omega ) = Z_{2}\left(\omega^{\prime }\right)$.

Complete the proof of the following lemma (Lemma 4). Let $Z_{1}: (\Omega , \mathcal{A}) \rightarrow \left(\Omega_{1}, \mathcal{A}_{1}^{\prime }\right)$ and $Z_{2}: (\Omega , \mathcal{A}) \rightarrow \left(\Omega_{2}, \mathcal{A}_{2}^{\prime }\right)$ be measurable, and suppose that the $\sigma$-field induced by $Z_{2}$, $\mathcal{A}_{2}$, satisfies $\mathcal{A}_{2} \subseteq \mathcal{A}_{1}$, the $\sigma$-field induced by $Z_{1}$. Then, if $\mathcal{A}_{2}^{\prime }$ contains the singletons of points in the range of $Z_{2}$, it follows that

1. $Z_{2}$ is a function of $Z_{1}$.

2. There exists a unique function $Z$ on $Z_{1}(\Omega )$, $\mathcal{A}_{1}^{\prime } \cap Z_{1}(\Omega )$ into $\left(\Omega_{2}, \mathcal{A}_{2}^{\prime }\right)$, defined by $Z\left(\omega_{1}\right) = Z_{2}\left(\omega ; Z_{1}(\omega ) = \omega_{1}\right)$, so that $Z_{2} = Z\left(Z_{1}\right)$.

3. $Z$ is $\mathcal{A}_{1}^{\prime } \cap Z_{1}(\Omega )$-measurable.

If $X_{n}, n \geq 1$, are i.i.d. r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$, then