Another equivalent notion of ergodicity

Question

Let $T\colon X\to X$ be a measure preserving transformation on a probability space $(X,\mu)$. We say that $T$ is ergodic if $A$ is measurable and $T^{-1}(A)=A$ implies that $\mu(A)\in\{0,1\}$. I have seen that the following are equivalent:

1. $T$ is ergodic,
2. If $B$ is measurable and $\mu(T^{-1}(B)\Delta B)=0$, then $\mu(B)\in\{0,1\}$. (Here $\Delta$ stands for the symmetric difference.)

However, I was reasoning as follows. If $C$ is measurable and $C\subset T^{-1}(C)$, then $$\mu(T^{-1}(C)\Delta C)=\mu(T^{-1}(C)\setminus C)+\mu(C\setminus T^{-1}(C))=\mu(T^{-1}(C)\setminus C)\\ =\mu(T^{-1}(C))-\mu(C)=\mu(C)-\mu(C)=0.$$ So I concluded that there was another characterization of ergodicity, namely:

3. If $C$ is measurable and $C\subset T^{-1}(C)$, then $\mu(C)\in\{0,1\}$.

Note that it is clearly indeed also true that 3 implies 1. But I find this characterization really counterintuitive, especially since I do not encounter this definition in the literature.

I think it is a nice characterization for the following reason: Suppose $T$ is ergodic and we want to show that a measurable set $C$ has either measure $0$ or $1$. Then, by 3, it suffices to prove that $C\subset T^{-1}(C)$. The reverse inclusion does not even have to hold!

So my questions are: Is my reasoning correct, that is, is $3$ indeed equivalent to ergodicity? Why does most literature (atleast the literature I found) choose not to mention anything about 3?

Answer 1

The notion (3) of ergodicity in the OP can be seen in terms of Markov chain terminology (that of absorbent sets or states). Suppose $(X,\mathscr{F})$ is a measurable space. Recall that a Markov transition probability function $P$ is a function $P:X\times \mathscr{F}\rightarrow[0,1]$ such that:

• For $x\in X$ fixed, $P(x,\cdot)$ is a probability measure on $(X,\mathscr{B})$.
• For $B\in \mathscr{F}$ fixed, $P(\cdot, B)$ is a measurable maps from $(X,\mathscr{F})$ into $([0,1],\mathscr{B}([0,1])$ (here $\mathscr{B}([0,1])$ refers to the Borel $\sigma$-algebra on $[0,1]$).

Notation:

• For $x\in X$, and $A\in\mathscr{F}$, we use $P\mathbb{1}_A(x):=P(x,A)$.
• More generally, for any nonnegative (or bounded) measurable function $f$ on $(X,\mathscr{B})$ we use $Pf(x):=\int f(y)\,P(x,dy)$.
• For any measure $\mu$ on $(X,\mathscr{B})$, define the measure $\mu P$ given as $\mu P(A)=\int P(x,A)\,\mu(dx)$, $A\in\mathscr{F}$.

Definitions

1. A set $A\in\mathscr{F}$ is absorbent if $P\mathbb{1}_A\geq \mathbb{1}_A$.
2. A measure $\mu$ in $(X,\mathscr{F})$ is $P$- invariant (or $P$ is $\mu$-invariant) if $\mu P(A):=\int P(x,A)\,\mu(dx)=\mu(A)$, $A\in\mathscr{F}$.
3. A set $A\in\mathscr{F}$ is $P$ invariant iff $P\mathbb{1}_A\stackrel{\text{$\mu$-a.s.}}{=}\mathbb{1}_A$, i.e. $\int|P\mathbb{1}_A-\mathbb{1}_A|\,d\mu=0$. Let $\mathcal{I}^\mu_P$ denote the collection of all $P$ invariant sets in $\mathscr{F}$.
4. A $P$-invariant measure $\mu$ is ergodic if $0\in\{\mu(A),\mu(X\setminus A)\}$ for any $A\in\mathscr{F}$.

Comments:

• (1) states that if the system $P$ starts at a point $x\in A$, then it will remain in $A$ ($P(x,A)=1)$.
• when $\mu$ is a $P$-invariant finite measure ( a probability measure for example), then ergodicity can be stated as $\mu(A)=0$ or $\mu(A)=\mu(X)$ for all $A\in\mathcal{I}^\mu_P$.

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The case of the OP can be identified with the Markov transition function $P$ defined as $P\mathbb{1}_A=\mathbb{1}_A\circ T$, that is $$P(x,A)=\mathbb{1}_A(T(x))=\mathbb{1}_{T^{-1}(A)}(x)$$

In this slightly more general setting,

• (i) a set $A$ is absorbent if $A\subset T^{-1}(A)$;
• (ii) a measure $\mu$ on $(X,\mathscr{B})$ is $P$ invariant iff $\mu(T^{-1}(A))=\mu(A)$ for all $A\in\mathscr{F}$, and
• (iii) a set $A$ is $P$-invariant iff $\mu(A\triangle T^{-1}(A))=0$.

It is not difficult to show that the standard definition of ergodicity for $\mu$-invariant transformation $T$ is equivalent to (iii) (regardless of whether $\mu$ is finite or not).

The notion of ergodicity that that the OP discovered is equivalent to stating that $T$-absorbent set has either measure $0$ or full measure.

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in the setting of Markov chains described above it is not difficult to show that

• (a) the collection $\mathcal{I}^\mu_P$ of invariant sets is a $\sigma$-algebra
• (b) If $A\in \mathcal{I}^\mu_P$ then there is an absorbent set $\hat{A}\in I^\mu_P$ such that $\mu(\hat{A}\triangle A)=0$
• (c) $P$ is $\mu$-ergodic (or $\mu$ is $P$-ergodic) iff $0\in\{\mu(A),\mu(X\setminus A)\}$ for any absorbent set $A\in \mathcal{I}^\mu_P$.
• (d) If $\mu$ is finite (a probability measure for example), then any absorbent set is also invariant, in which case, $P$ is $\mu$ ergodic iff $\mu(A)=0$ or $\mu(A)=\mu(X)$ whenever $A$ is absorbent.

Remark: The equivalence (c) in the slightly more general setting of Markov chains is what the OP discovered in the case of invariant maps.

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There are several treatments of the Ergodic theory for Markov chains:

• the book of Meyn, S. and Tweedie, R.L., Markov Chains and Stochastic Stability, Second editions, Cambridge is an excellent reference;
• the book of Krengel, U., Ergodic Theorems, de Gruyter Studies in Mathematics, contains a treatment of the ergodic theory of Markov chains.
• the book Kallenberg, O., Foundations of Modern Probability, second edition, Springer-Verlag, has a chapter on the ergodic theory of positive contractions that covers Markov chains (and the particular case of invariant transformations).