# An introduction to ergodic theory - Walters P.

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$0.6 Ha;ir .Measure

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Proof. For/e Ll(X,68,m) we know £(//#) is the only 'if-mea.jurable function Ii such that \chdm = Jc/dm VC e 'й7. Let P denote the orthogonal projection of L2(X, HS, m) onto the closed subspace L2(X,V,tn). If/ e L2(X,3S,m) then Pf is 'iS’-mcasurable and if С e V

\cf dm = (f,yj = (f,P7J = (P/,Zc) = Jc Pf dm.

Therefore Pf = E(f/V). □

§0.6 Haar Measure

There is a probability measure on a compact group G which ties in with the group structure on G. This measure is defined on the er-algebra 36(G) of all Borcl subsets of G. It will also have the property of regularity. Recall that a measure m on the Borcl (7-algebra a6(X) of a compact topological space X is regular if for every i: > 0 and every £ l ЩХ) there is a compact set M and an open set U with M с E <= U and m(U\M) < e. It suffices to require that for each e > 0 and E e 36(X) there is a compact set M with M cz E and ni(E\M) < r. (since we can also apply this to X\E to get an open set U with E cz U and m{U\E) < c). If X is metnsable then any probability measure on (X,J6(X)) is regular (see Theorem 6.1).

Theorem 0.13. Let G be a compact topological group. There exists и probability measure m defined on the a-algebra 36(G) of Borel subsets of G such that m(.\E) = m(E) V.x e G V£ e 36(G) and m is regular. There is only one regular rotation invariant probability measure on (G, 36(G)).

This unique measure is called Haar measure. Notice we have required llaar measure to be a probability measure. The Haar measure m also satisfies in(Ex) = ж(£) Vx e G, V£ e :t6(G), because for each fixed x e G the measure mx defined by mx(E) = m(Ex) is rotation invariant and regular and hence equals m.UG is metrisable then, as mentioned above, any probability measure on (C,36(G)) is regular so we can omit the regularity assumption from the statement about uniqueness of Haar measure.

The rotation invariance of m can also be expressed by requiring \с!(хУ) dm(y) = jo f(y)dm(y) V/ e L\m\ Vx e G.

If U is a non-empty open subset of G then it has non-zero Haar measure, because G — U,;6C gU = gyU и g2U u • • • u gkU by compactness.

For the circle group К — {z e C| |z| = 1} the Haar measure is the normalised circular Lebcsgue measure. For the н-torus K" the Haar measure is the direct product of the Haar measure on K.

If m; is the Haar measure on Git ieZ, then the direct product of the measures mf is the Haar measure on the direct product group G,.

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0 Preliminaries

For the two point group (0,1} the Haar measure gives cach point measure i so the I la.ir measure on the direct product group ]”fjt {0,1} is the dnect product of this measure.

On anv compact metrisable group G there is a metric p which is rotation invariant in the sense that pU)X,(jy) = p(x,y) = p(xiy, ytj) V#, x, у e G. If (I is any metric on (', and m is I laar measure wc could take p(x, y) = gyh)

tlmlij) )ilin(ii).

§0.7 Character Theory

Many of our examples will be rotations, endomorphisms, or affine transformations of compact groups. (Wc mean endomorphism in the sense of topological groups, i.e., an abstract group endomorphism which is continuous.) In some proofs wc will use the character theory of compact abelian groups which we summarise in this section. For those not familiar with character theory, proofs in the later sections involving character will usually be preceded by the proof in a special case where the group used is the unit circle and then classical Fourier analysis will be used The proofs of all results quoted in this section can be found in Hewitt and Ross [1].

Let G be a locally compact abelian group. Let G denote the collection of all continuous homomorphisms of G into the unit circle K. The members of G are the diameters of G Under the operation of pointwise multiplication of functions (i is an abelian group. With the compact open topology G bccomcs a locally compact abelian group.

In §0.N we shall show that when G — К = (ее C| |zj = 1} each element of К is of the form z -> z" for some n e Z. Hence К ^ Z. We also show that the character group of the n-torus K" is isomorphic to Z" and each у e K" is of the form

y(z......= -i\ -2*.....=n” for some (p,,p2,... ,p„) e Z".

We have the following results.

(1) G has a countable topological basis iff С has a countable topological basis

(2) G is compact ifT G is discrete.

Combining (2) with (1) wc have G is compact and metrisable iff G is a discrete countable group. This allows us to transform some problems about compact abelian metrisable groups to problems about discrete countable abelian groups.

(3) (Duality Theorem). (G) is naturally isomorphic (as a topological group) to G, the isomorphism being jiven by the map a -* a where a(y) = y(a) for all у e G.

jjO 7 Character Theory

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(4) If С is compact then G is connected iff 6 is torsion free (i.e. has no elements of finite order apart from the identity element.)

p) If G,, G2 are locally compact abelian groups then G, xC2 = C, x C.. (Here “ x " denotes direct product.) Hence all characters of G2 x G2 are of the form (x, y) -* y(x)S(y) where у eGlt6e G2.

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