# An introduction to ergodic theory - Walters P.

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m(Uc\L\) < m(U\Cc) + m{Cc\Cr)

< Z >n(Uc.n\CeJ + ш(СДС£)

n = L * £ f. n?i 3" +2 - £*

Therefore 'A is a a-algebra.

To complete the proof we show that Si contains all the closed subsets of A. Let С be a closed set and £ > 0. Define Un = {x e A": J(C. л | < I /и}. This is an open set, 2 l/2 2 - • • 2 Lr„ 2 ■ ■ • and у Ui — C. Choosc к such that m(Uk\C) < г and let C£ = С and Uc = Uk. This shows С e J) □

Corollary 6.1.1. Fora Borel probability measure топа metric spacc A we I шее that for Be ЩХ)

m(B)= sup m(C), and m(B) — inf m{U).

С closed V open

сев изо

The next result says that each m e M(A) is determined by how it integrates continuous functions.

Theorem 6.2. Let m, ц be two Borel prooaniliiy measures on the metric space X. Iffx fdm = [xf dp. V/ 6 С(A) then m = fi.

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6 Invariant Measures foi Continuous Transformations

Pjuxik By the above corollary it sufTiccs to show that »жС) = //(С) for all closed set' (' ~ X. Suppose С i1 do ctl and let .. "> 0. iiy the rcg-jl.irity of m 1 Р|£гТГс1«ч^цщ_о£1Сn >et и w.ih ( rz L‘ and nili C) < «

Define J: X -» R "By—' ——

(0 ir a- s U

_‘l(x,X_U) dix.X U) + d(x,C)

This is well-defined since the denominator is not /его. Also / is continuous, / = 0 on X \L', / = 1 on C, and 0 < f(x) < 1 Vx e .Y. Hence

p(C) < Jx fdp = JA, /dm < m(U) < мцС) + г..

Therefore p(C) < m(C) + e Ve > 0, so p(C) < m(C). By symmetry we get that m(C) < /((C). □

The next theorem relates elements of M(X) to linear functionals on C(X).

Theorem 6.3 (Riesz Representation Theorem). Lei X be a compac metric space and J :C(X) С a continuous linear map such that J is a positive operator (i.e., if f > 0 then J(f) > 0) and J( 1) = 1. Then there exists p e M(X) such that Jif) = UfdnVfeC( X).

For the proof see Parthasarathy [2] p. 145.

Therefore the map p -»J is a bijection between M(X) and the collection of all normalised positive linear functionals on C\.Y). (Injectivity follows from Theorem 6.2 and surjectmty by Theorem 6.3.) We shall denote the image of p. under this map by JClearly this bijection is an affine inrfrp (i.t\ J + (, p)„ — pj„ + (1 — p)Jm, p e [0,1 J, ж, ц e M(X)) so M(X) is identified with a convex sublet of the unit ball in C(.Y)* This allows us to gel a topology on M(X) from the weak* topology on C(.Y)*.

Definition 6.1. The weak* topology on M(X) is the smallest topology making cach of the maps p~*Jx /dp (f e C(X)) continuous. A basis is given by the collection of all sets of the form VJ f^ ... ,fk; z) = {in e M(X) \ \\fdm — §fidfi\ < £, 1 < i < k} wnere p e M(X), к > 1, f e C(X) and e > 0.

Clearly this topology on M(X) is independent of any metric chosen on X.

Theorem 6.4. If X is a compact metrisable space then the space M(X) is metrisable in the weak* topology If {/,}*= i is a dense subset of C(X) then

is a metric on M(X) giving the weak* topology.

^6.1 Measures on Mctric Spaces

149

Proof. The function D:M(X) x R is clearly a metric. Consider the

metric space (M(X), D). For each fixed i the map f t -*jf dfi is clearly con-

dense in C(X) it follows that for each / e C(X) the map -♦ j/d/i is continuous on (M{X), D). Therefore-every upeii set in the wc.il * topology is open in the metric space (\UX),D). To show the converse it will suffice to prove each ball {(it e M(X)\D(m,fi) < e} in (M(X), D) contains a set V,Uh, ■ ■ ■ ,9*; 6) where k > 1, e C(X) 1 < i < к, and о > 0 If ^ e M(X) and e > 0 are given, choose N so that

(1) In the weak* topology /г„ -* ц in M[X) iff V/ e C{X) -*\f dfi.

(2) The imbedding X -» M(X) given by x -»is continuous.

(3) If ц„, ц e M(X), n> 1, one can prove the following aie equivalent.

(i) /t„ -»ц in the weak*-topology

(ii) For each closed subset F of X, lim sup ^„(F) < /<(F).

(iv) For every A e af with /*(<3.4) = 0, /i„(/l) -* ц(Л).

We shall want to use (i) => (iv) so we give a proof. Wc shall in fact show that (i) => (ii) and then (ii) => (iii) and (iv). Let F be a closed subset of A' and for к > 1 let Uk = {* e X\cl(x, F) < l/k}. The sets Uk are open and decrease to F. Therefore n(Uk) -* n(F). By Urysohn’s lemma choose fke C(Ar) with 0 < fk < 1 ,fk= 1 on F and /* = 0 on X\Uk. Then

so lim sup„_ ,, n„(F) < fi(F). Therefore (i) => (ii). Suppose (ii) is true and let U be an open subset of X. Then

lim sup n„{X\U) < n(X\U) so lim inf n„(U) > n(U).

Therefore (ii) => (iii). Suppose (ii) is true and ц(дА) = 0. Then /<(int(y4)) = ц{А) = ц(А) and lim sup„_ ц„(А) < ц(А) = ^(/l)and lim inf^^ /i„(int(A))> j/(int(/l)) = ц{А). Therefore ц„(А) -»ц(Л). We have shown (i) => (iv). □

tinuous on (X,D) because — $fdp\ < 2'||_/j||Z)(»^/i). Since {f;} f is

n = n +1

I

Let

Then VJLf\.....fN; 3) c= {m e M{X)\D{m.fi) < e}.

□

Remarks

(iii) For each open subset U of A", lim inf > n(U).

Iii = ^hi<fi(Uk)

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6 Invariant Measures lor CoMmuous Transtorir.jiions

(See Parthasarathy [1] for the proofs of the other implications.)

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