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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 .. 99 >> Next Let (X,3d, ui) be a measure space. A function /:X ‚Äî‚ñ∫ R is measurable if f~ \D) e 3 whenever D e .H(R) or equivalently if/- ‚Äò(—Å, —Å–æ) e ^ for all ce R. A function f: X -* –° 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 —Å—Ç-algebra generated by the open subsets of A", then any continuous function f:X -> –° is measurable. We say / = g a.e. if m({x:/(x) —Ñ g(x)j) = 0. Suppose A is a topological space, .–Ø‚Äô(–ê') its tx-algcbra of Bore) sets and m a measure on (–•,–©–•)) 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) ‚Äî g(x) —Ñ 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 –∞^/–¥., where a; e/i, /);e the sets Aj are disjoint subsets of A‚Äô, 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‚Äû_a J/, An and note that this definition is independent of the chosen sequence (/,}. We say / is integrable if J /dm < —Å–æ.
Suppose f:X-*R is measurable. Then / = /+ ‚Äî f~ where f+{x) = max {/(*), 0} ;> 0 and f~ (x) = max {‚Äî/(x), 0} > 0. We say that / is integrable if j/ + dm, J f~ dm < —Å–æ and we then define
J/An = J7+An - J7~ An.
We say /: X -* –° 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 <
‚Ä¢ ‚ñÝ ‚Ä¢ 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‚Äû_ 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‚Äû_a /‚Äû is infinite on a set of positive measure or lim,,..,*, fn is not integrable.
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0 Preliminaries
Theorem 0.9 (Fatou‚Äôl Lemma). Let { [,} be a sequence of measurable real-valued functions on ( V, ‚ñÝ/), in) winch is bounded below by an intcgrable function. If lim i 11 f‚Äû_ , J f,din < oj then Jim inf‚Äû^,y f, is intcgrable and J lim \ni f, dm < liminf \f,dm.
Corollarj 0.9.1 (Dominated Convergence Theorem). If cj:X -¬ªR is inte-grable and {[,)‚ñÝ is a sequence of measurable real-valued functions with \ f\ < —É a.e. (n > 1) and lim,,., u f, ‚Äî 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‚Äô -> –° where two such functions are identified if they are equal a.e. However wc write f e Ll(–•,3–≤,—Ç) to denote that/: X -* –° 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 ‚ñÝ /Adm.
If m is a finite signed measure on (X,.yJ) and m = ;?i, ‚Äî m2 is its unique Jordan decomposition into the dillerencc of two finite measures, then we can define j/dm ‚Äî jf d¬ªiI ‚Äî J/dm2 for /e Ll\tni) –≥–ª Li(m2).
¬ß0.4 Absolutely Continuous Measures and Conditional Expeditions
Let (–•,.–© be a measurable space and suppose /(, m are two probability measures on (–ê', –õ). We say —Ü is absolutely continuous with respect to —Ç(/–ª ¬´ m) if —Ü(–í) = 0 whenever in(B) = 0. The measures are equivalent if —Ü ¬´ 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 —Ü(–í) = J –π f din VB e –õ. 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 —Ü with respect to m and denoted by du/dm.
The ‚Äúopposite‚Äù notion to absolute continuity is as follows. Two probability measures —Ü, m on (X,S6) are said to be mutually singular (—Ü _L m) if there is some –í ‚Ç28 with —Ü(–í) = 0 and in(X\B) = 0. There is the following decomposition theorem.
Theorem 0.11 (Lcbesguc Decomposition Theorem). Let —Ü,—Ç be two probability measures on (A', .iS). There exists p e [0,1] and probability measures —Ü–∏ —Ü2 on (X, 3/1) ‚ñÝsuch that // = —Ä/1{ -I- (1 ‚Äî —Ä)—Ü2 and ¬´ m, fi2 -L m. (fi = pp\ + (1 ‚Äî p)jl2 means /i(B) - p/i^lS) + (1 ‚Äî —Ä)—Ü2(–í)–£–í e S). The number p and probabilities fi,, fi2 are uniquely deiemiinad.
¬ß0.5 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): -> Previous << 1 .. 2 3 < 4 > 5 6 7 8 9 10 .. 99 >> Next 