# An introduction to ergodic theory - Walters P.

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The measure m induces a measure ffx on 36 by m(B) = m(B). (Here В is the equivalence class to which В belongs.) The pair [$, in) is called a measure algebra.

From this point of view one says (Xl,^l,m1) and (X2,3t2,m2) are “equivalent” if their corresponding measure algebras are isomorphic:

Definition 2.2. Let (Xt, &т^, {X2,3S2,m2) be probability spaces with measure algebras (3S2,ih2)- The measure algebras are isomorphic

if there is a bijection Ф:382 -» ^ which preserves complements, countable unions and intersections and satisfies т^ФВ) = m2(B) VZ? e 3S2. The probability spaces are said to be conjugate if their measure algebras are isomorphic

Conjugacy of measure spaces is weaker than isomorphism because if (Xar.d (X2,3d2,m2) are isomorphic as in Definition 2.1 then they are conjugate via Ф:3&2 defined by Ф(В) = (ф~1(М2 n B))~. It is easy to give examples of conjugate measure spaces which are not isomorphic. Let (Л-!, $?,,»?,) be a space of one point and let {X2,332,m2) be a space with two points and 362 = {ф,Х2}. The two spaces are conjugate but they are not isomorphic because a set of zero measure cannot be omitted from X 2 so that the remaining set is mapped bijectively with Xx. The main reason this example works is that -.JA2 does not separate the points of X2. When some conditions are placed on the probaoility spaces the notions of conjugacy and isomorphism coincide:

$2.1 Point Maps and Srt Maps

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Theorem 2.2. Let XltX2 be complete separable metric spaces, let 36 (X^), ^(X2) be their a-algebras of Borel subsets and let m,, m2 be probability measures on Sd(X {), J}(X2). Let Ф:3&(Х2) —* fti(Xx) be an isomorphism of measure algebras. Then there exist M, e 36(Xx), M2 e 3H(X2) with —

1 = m2(M2) and an invertible measure-preserving transformation ф:М l —► M2 such that Ф(В) = (ф~'(В n M2)) VB e 3S(X2). If Ф is any other isomorphism from {Х1,ЩХ1),/«!) to (X2,ЩХ2),m2) which induces Ф then

т,({л еХ^фМ ф i]/(x)}) = 0.

The proof is given in Theorem 12, page 329 of Royden [1].

The corresponding statement for measure-algebra homomorphisms holds (they are induced by (not necessarily invertible) measure-preserving transformations).

Therefore set maps are always induced by point maps for the probability spaces used in our examples.

Often in the ergodic theory literature all probability spaces used are assumed to be Lebesgue spaces •

Definition 2.3. A probability space (X, 3, m) is a Lebesgue space if it is isomorphic to a probability space which is the disjoint union of a countable (or finite) number of points y2, ■ ■ •} each of positive measure and the space ([0,5], JS?([0,s],l) where Jz?([0,s]) is the ег-algebra of Lebesgue measurable subsets of [0, s] and I is Lebesgue measure. Here 5=1— 1 p„ where pn is the measure of the point y„.

The theory of Lebesgue spaces is given in Rohlin [1]. In particular the analogue of Theorem 2.2 is true for Lebesgue spaces (i.e. set maps are always induced by point maps) and so the two ways of dealing with sets of measure zero coincide. Notice that the remarks following Theorem 2.1 show that if X is a complete separable metric space and zdm(X) denotes the completion оШ{Х) under a probability measure m then (X, B6m(X), m) is a Lebesgue space.

A third way to study a probability space (X,£8,m) is to study the Hilbert space L2(m). If {Xu88umy), (X2,&2,m2) are both spaces with countable basis then £2(тД L2(m2) are separable (see §0.5) and hence unitarily isomorphic (i.e. there is a bijective linear map W:L2(m2) -*■ L2(mf) such that [Wf, Wg) = (f,g) V/, g e L2{m2)). However, L2(m) has some extra structure because one can multiply certain members of L2(m) to obtain a function in L2(m). It turns out that conjugacy of measure spaces is equivalent to their L2 spaces being equivalent in a sense involving this multiplication.

Let (X, 38, m) be a probability space. Since L2(m) consists of equivalence classes of functions (/ and g are equivalent if / = g a.e.) we see that if BeJ then xb is a well-defined member of L2(m). There is the following simple result.

Theorem 2.3. Let (Xl,3d1,m1), (X2,3ti2,m2) be probability spaces and let Ф:(ЗЯ2,т2) -* (3S2,inj) be a measure algebra isomorphism. Then Ф induces a

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’ Isomorphism. Conjugacy. and Spectral Isomorphism

bijcctivc linear map \':L2(m2) —* L2(»i;) with the properties

(a) (IУд) = (f,g) V/, g e L2(m2)

(b) V and V~1 map bounded functions to bounded functions

(c) = (КЛ( ^g) whenever f, g are bounded.

The тар V is defined on characteristic functions by Ууц = уа,(щ.

Proof. For a characteristic function уц, Be 32 define Ууц = Уффу Notice II^ZbIU = ilZ/ilU- Extend V to simple functions to preserve linearity i.e. ^(£'=i UiXit) = Yj = 1 a, VXи, whenever {B,,..., B„} are mutually disjoint members of j$2. This definition is independent of the representation of a simple function as a linear combination of characteristic functions. We have ЦК/Ц2 = ll/lb f°r aM simple functions /. Now let / be any element of L2(m2). Choose simple functions f„ with Ц/, — /||2 ->• 0. Since {/„} is a Cauchy sequence and V is an isometry on simple functions we know {Vfn} is a Cauchy sequence. Denote the limit of this sequence by Vf. If {g„} is another sequence of simple functions with ||g„ — /||2 -* 0 then ||g„ — /„||2 -» 0 so that IlK/ii- Kg„||2 —>• 0. This shows that Vf is well defined. The inverse of V is constructed in a similar way from Ф-1. The linearity of V is clear from the definition. Property (a) holds because it is equivalent to V preserving norm. A function / e L2(m2) is bounded iff it is the limit of a sequence of simple functions which are uniformly bounded. By the definition of V on simple functions we see that Vf has the same bounds as /. Property (c) is easily

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