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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 2 3 4 < 5 > 6 7 8 9 10 11 .. 99 >> Next L'(A\e¬£,m). If/eL'(X,34,in) lakes non-negative real values then nf(C) = <–ì1 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,‚Äò) > 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(–•/<?,—Ç) hold (Parthasarathy , p. 225):
(i) ¬£(-fe) is linear.
(ii) ir/^0, then ¬£(//¬´‚Äô) > 0.
(iii) If/e Ll(X,3S,m) and g is *if-measurable and bounded,
EifgM) = gE(f/n
(iv) –©.—Ç\ < ¬£( \–®), f 6 L\X,a,m)
(v) l(V2 —Å <eu then EiEif/VM‚Ç2) = ¬£(/A?2), / e L\X, 36, m).
¬ß0.5 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 (–•,9–≤,—Ç) be a measure space and let p e R with p > 1. Consider the set of all measurable functions f:X -* –° 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 -> –° has |/|p integrable. The formula jj/||,, = [J‚Äô|/|,‚Äôi/¬ªi]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, –õ, 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 –Ω—Ü–ê") < —Å–æ and I < p < q then L*(X,J&,m)cz –—Ä(–•,2–≤,—Ç). We sometimes write Lp(in) or Ln(.-^) instead of Lp(X,&l,m) when no confusion can arise.
–õ Hilbert space –ñ is a Banach space in which the norm is given by an inner product, i.e., –ñ is a Banach space and there is a map (‚Ä¢, ‚Ä¢): –ñ x –ñ –° such that (‚Ä¢, ‚Ä¢) is bilinear, (f,g) = (g,f) Vg,f 6 –ñ, (/, /) > 0 V/ e –ñ\ and / = (/,/)1/2 is the norm on –ñ.
–Æ 0 Prclimin;
The –ü–∞–ø–∞—Å– space Lp(X..'J4,n}) is a Hilbert space iff p = 2. The inner
‚Ä¢ pttxlcci in is ‚Äùivcn 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 –î ¬£‚Äû) < 'if A' is a metric space and /j is the –≥–≥-algebra of Borel subsets of X (the <7-algebra generated by the open sets) and in is any probability measure on (iA\ –© 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 –ñ contains a basis \c‚Äû) J, i.e. ek) = 0 if n —Ñ –∫ and only the /–µ–≥–æ clement is orthogonal to all the e‚Äû. If is a basis then each v e –ñ is uniquely expressible as v = j a‚Äûe‚Äû where a‚Äû e C. We have
ll'il2 = Z –ò2 so lhat Z –´2 <oc-
n=i ¬ª=i
An isomorphism between two Hilbert spaces –ñu –ñ–≥ is a linear bijcction W: '//x -‚ñ∫ :/(\ that preserves norms (–¶–ú^-–¶ = ||i'|| Vi> e –ñ\). The norm-preserving condition can also be written as [Wu, Wi) ¬´= [u, v) Vu, –∏ 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 –∫ 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 /–ì then V1 = {h e |(u, h) = 0 Vu e V) is a closed subspace of –£/ and V ¬© V1 = –ñ (i.e. each f e –ñ has a unique representation / =fi+ /2 where /, e Kand/2 e VL.) The linear operator P: –ñ -* V given by P(f) = fx is called the orthogonal projection of –ñ onto V. In fact P(f) is the unique member of V that satisfies ||/‚Äî ¬£(/)|| = inf {||/ ‚Äî y|| I v e K}. We have P\V = id and (Pf.g) = (/, Pg) V/, g e –ñ.
Let (–ê', ?J), in) be a probability space and recall from ¬ß0.4 that if V is a sub-rr-algebra of .¬£ then the conditional expectation operator  