# An introduction to ergodic theory - Walters P.

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L'(A\e£,m). If/eL'(X,34,in) lakes non-negative real values then nf(C) = <Г1 jef dm (where a = \xf dm) defines a probability measue, /(y, on {X/6\ in) and nf « m. By Theorem 0.10 there is a function E(f/%J) e L1(X,^,m) such that E{f/Y,‘) > 0 and J*c E(fK)dm = Jc/dmVC e CG. Moreover £(//"6) is unique a.e. If / is real-valued we can consider the positive and negative parls of / and define E{ffC) linearly. Similarly when / is complex-valued we can use the real and imaginary parts to define E(f/'o) linearly. Therefore if / 6 Ll{X,.i6,m) ihen E{fjrfi) is the only 'if-measurable function with \cEW)d,n = IcfdmVCeV. The following properties of the map E(-/W):Ll(X, £,m) -* 1(Х/<?,т) hold (Parthasarathy [2], p. 225):

(i) £(-fe) is linear.

(ii) ir/^0, then £(//«’) > 0.

(iii) If/e Ll(X,3S,m) and g is *if-measurable and bounded,

EifgM) = gE(f/n

(iv) Щ.т\ < £( \Ш), f 6 L\X,a,m)

(v) l(V2 с <eu then EiEif/VM2) = £(/A?2), / e L\X, 36, m).

§0.5 Function Spaces

One way to deal with some problems on a measure space is to use certain natural Banach spaces of functions associated with the measure space.

Let (Х,9в,т) be a measure space and let p e R with p > 1. Consider the set of all measurable functions f:X -* С with \ f\p integrable. This space is a vector space under the usual addition and scalar multiplication of functions. If we define an equivalence relation on this set by / ~ g iff / = g a.e. then the space of equivalence classes is also a vector space. Let Lp(X,.Ji,m) denote the space of equivalence classes, although we write / £ LP(X, in) to denote that the function /: X -> С has |/|p integrable. The formula jj/||,, = [J’|/|,’i/»i]1/,,delnesa norm on Lp{X,36,m) and this norm is complete. Therefore Lp(X,34,m) is a Banach space. If L','((X, Л, in) denotes those equivalence classes containing real-valued functions then Lp{{X,3ti,in) is a real Banach space. The bounded measurable functions are dense in Lp(X, JS, m). If нцА") < со and I < p < q then L*(X,J&,m)cz р(Х,2в,т). We sometimes write Lp(in) or Ln(.-^) instead of Lp(X,&l,m) when no confusion can arise.

Л Hilbert space Ж is a Banach space in which the norm is given by an inner product, i.e., Ж is a Banach space and there is a map (•, •): Ж x Ж С such that (•, •) is bilinear, (f,g) = (g,f) Vg,f 6 Ж, (/, /) > 0 V/ e Ж\ and / = (/,/)1/2 is the norm on Ж.

Ю 0 Prclimin;

The Папас space Lp(X..'J4,n}) is a Hilbert space iff p = 2. The inner

• pttxlcci in is ”ivcn by If.u) = | fg din.

In c\ery Hilbert .space.// wc have the Schwarz inequality:

<|/|- jj./i! Vf.ge.//*.

Separable Hilbert spaces (i.e. those having a countable dense set) are the simplest. The space l.:,X, m) is separable iiT(A'. /1, in) has a cnuniable basis, in the sense that there is a sequence of elements [£,,}? of M such that for every i: > 0 and every Be.^ with m(/i) < x there is some n with in(B Д £„) < 'if A' is a metric space and /j is the гг-algebra of Borel subsets of X (the <7-algebra generated by the open sets) and in is any probability measure on (iA\ Щ then (X, rtf,in) has a countable basis. (This follows from Theorem 6.1.) Therefore most of the spaces wc shall deal with have L2(Xm) separable.

Any separate Hilbert space Ж contains a basis \c„) J, i.e. ek) = 0 if n ф к and only the /его clement is orthogonal to all the e„. If is a basis then each v e Ж is uniquely expressible as v = j a„e„ where a„ e C. We have

ll'il2 = Z И2 so lhat Z Ы2 <oc-

n=i »=i

An isomorphism between two Hilbert spaces Жu Жг is a linear bijcction W: '//x -► :/(\ that preserves norms (ЦМ^-Ц = ||i'|| Vi> e Ж\). The norm-preserving condition can also be written as [Wu, Wi) «= [u, v) Vu, и e Any two separable Hilbert spaces are isomorphic if they both have a basis with an infinite number of elements. A Hilbert space with a basis of к elements is isomorphic to C*. An isomorphism of a Hilbert space "/C to itself is called a unitary operator.

If V is a elosed subspace of a Hilbert spacc /Г then V1 = {h e |(u, h) = 0 Vu e V) is a closed subspace of У/ and V © V1 = Ж (i.e. each f e Ж has a unique representation / =fi+ /2 where /, e Kand/2 e VL.) The linear operator P: Ж -* V given by P(f) = fx is called the orthogonal projection of Ж onto V. In fact P(f) is the unique member of V that satisfies ||/— £(/)|| = inf {||/ — y|| I v e K}. We have P\V = id and (Pf.g) = (/, Pg) V/, g e Ж.

Let (А', ?J), in) be a probability space and recall from §0.4 that if V is a sub-rr-algebra of .£ then the conditional expectation operator

Е(-1Щ:0(Х,3$,т) - L\X,V,m)

is defined. Since L2(X\9&,m) с Ll(X,j$,m) the conditional expectation operator acts on L2(Ar.,3t,m) and the following result describes what this restriction is.

Theorem 0.12. Let (X, 36, m) be a probability spacc and let (C be a sub-o-algebra of 66. The restriction of the conditional expectation operator £( ■/%’) to L2(X, 36, m) is the orthogonal projection of L2(X, 36, in) onto L2(X, (S, in).

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