# An introduction to ergodic theory - Walters P.

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Let (X,3d, ui) be a measure space. A function /:X —► R is measurable if f~ \D) e 3 whenever D e .H(R) or equivalently if/- ‘(с, со) e ^ for all ce R. A function f: X -* С i> measurable if both its real and imaginary parts are

jfO.3 Integration

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measurable. If A' is a topological space and the ст-algebra generated by the open subsets of A", then any continuous function f:X -> С is measurable. We say / = g a.e. if m({x:/(x) ф g(x)j) = 0. Suppose A is a topological space, .Я’(А') its tx-algcbra of Bore) sets and m a measure on (Х,ЩХ)) with the property that each non-empty open set has non-zero measure. Then for two continuous functions f,g:X->R,f*=g a.e. implies/ = g because {x:/(x) — g(x) ф 0} is an open set of zero measure.

Let (X,$S,m) be a probability space. A function f:X-*R is a simple function if it can be written in the form £"= t а^/д., where a; e/i, /);e the sets Aj are disjoint subsets of A’, and %Al denotes the characteristic function of A;. Simple functions are measurable. We define the integral for simple functions by:

J/Ah = £ a.wM,).

This value is independent of the representation

Suppose/: X -* R is measurable and/ > 0. Then there exists an increasing sequence of simple functions fn s f. For example, we could take

fkA =

'Jn 1 *yn - J v * 'yn

n, if f(x) > n.

We define \f dm = lim„_a J/, An and note that this definition is independent of the chosen sequence (/,}. We say / is integrable if J /dm < со.

Suppose f:X-*R is measurable. Then / = /+ — f~ where f+{x) = max {/(*), 0} ;> 0 and f~ (x) = max {—/(x), 0} > 0. We say that / is integrable if j/ + dm, J f~ dm < со and we then define

J/An = J7+An - J7~ An.

We say /: X -* С is integrable (/ = /t -t- if 2) if/i andf2 are integrable and we define

J/ dm = J/j dm + i J/2 dm.

Observe that f is integrable if and only if |/| is integrable. If / = g a.e. then one is integrable if the other is and J/ dm = jgdm.

The two basic theorems on integrating sequences as functions are the following.

Theorem 0.8 (Monotone Convergence Theorem). Suppose /, </. </3 <

• ■ • is an increasing sequence of integrable real-valued functions on (X,9$,m). If {J7« dm} is a bounded sequence of real numbers then lim„_ x fn exists a.e. and is integrable and J(hm f,)dm = lim Jfn dm. If {|/, dm) is an unbounded sequence then either lim„_a /„ is infinite on a set of positive measure or lim,,..,*, fn is not integrable.

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0 Preliminaries

Theorem 0.9 (Fatou’l Lemma). Let { [,} be a sequence of measurable real-valued functions on ( V, ■/), in) winch is bounded below by an intcgrable function. If lim i 11 f„_ , J f,din < oj then Jim inf„^,y f, is intcgrable and J lim \ni f, dm < liminf \f,dm.

Corollarj 0.9.1 (Dominated Convergence Theorem). If cj:X -»R is inte-grable and {[,)■ is a sequence of measurable real-valued functions with \ f\ < у a.e. (n > 1) and lim,,., u f, — f a.e. then f is intcgrable and lim J f, dm = J/dm.

Wc denote by L'(X,ri,ni) (or L'{in}) lliu SpJLe of all integrable functions / A’ -> С where two such functions are identified if they are equal a.e. However wc write f e Ll(Х,3в,т) to denote that/: X -* С is intcgrable. The space Ll[X,30,m) is a Banach space with norm |j/]|, = yf\dm.

If/e Ll(X,.^,in), then \Af dm denotes \ f ■ /Adm.

If m is a finite signed measure on (X,.yJ) and m = ;?i, — m2 is its unique Jordan decomposition into the dillerencc of two finite measures, then we can define j/dm — jf d»iI — J/dm2 for /e Ll\tni) гл Li(m2).

§0.4 Absolutely Continuous Measures and Conditional Expeditions

Let (Х,.Щ be a measurable space and suppose /(, m are two probability measures on (А', Л). We say ц is absolutely continuous with respect to т(/л « m) if ц(В) = 0 whenever in(B) = 0. The measures are equivalent if ц « m and ni «/I. The following theorem characterises absolute continuity.

Theorem 0.10 (Radon-Nikodym Theorem). Lei p, m be two probability measure on the measurable space (X, Jd). Then ft « in iff there exists f e L'(m), with f ;> O a/u/Jfdm = 1, such that ц(В) = J й f din VB e Л. Thefunction f is unique a.e. (in the sense that any other function with these properties is equal to f a.e.).

The function / is called the Radon-Nikodym derivative of ц with respect to m and denoted by du/dm.

The “opposite” notion to absolute continuity is as follows. Two probability measures ц, m on (X,S6) are said to be mutually singular (ц _L m) if there is some В 28 with ц(В) = 0 and in(X\B) = 0. There is the following decomposition theorem.

Theorem 0.11 (Lcbesguc Decomposition Theorem). Let ц,т be two probability measures on (A', .iS). There exists p e [0,1] and probability measures ци ц2 on (X, 3/1) ■such that // = р/1{ -I- (1 — р)ц2 and « m, fi2 -L m. (fi = pp\ + (1 — p)jl2 means /i(B) - p/i^lS) + (1 — р)ц2(В)УВ e S). The number p and probabilities fi,, fi2 are uniquely deiemiinad.

§0.5 Function Spaccs

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The Radon-Nikodym theorem allows us to define conditional expectations. Let (X\3S,m) be a measure space and let c6 be a sub-c-algebra of SS We now define the conditional expectation operator E(-/V): ->

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