Appendix A. Some Topology and Measure Theory (i) Topology
The concept of a topological space is so fundamental in modern mathematics that we don’t feel obliged to recall its definitions or basic properties. Therefore we refer to Dugundji 1966 for everything concerning topology, nevertheless we shall briefly quote some results on compact and metric spaces which we use frequently.
A.1. Compactness:
A topological spacepX,Oq,Othe family of open sets inX, is calledcompact if it is Hausdorff and if every open cover ofX has a finite subcover. The second property is equivalent to the finite intersection property: every family of closed subsets of X, every finite subfamily of which has non-empty intersection, has itself non-empty intersection.
A.2. The continuous image of a compact space is compact if it is Hausdorff. More- over, if X is compact, a mappingϕ:X ÑX is already a homeomorphism if it is continuous and bijective. IfX is compact for some topologyO and ifO1 is another topology onX, coarser thanO but still Hausdorff, thenOO1.
A.3. Product spaces:
Let pXαqαPA a non-empty family of non-empty topological spaces. The product X :±
αPAXαbecomes a topological space if we construct a topology onX start- ing with the base of open rectangles, i.e. with sets of the form tx pxαqαPA : xαi POαi fori1, . . . , nuforα1, . . . , αn PA, nPN andOαi open in Xαi. Then Tychonov’s theorem asserts that for this topology,X is compact if and only if each Xα,αPAis compact.
A.4. Urysohn’s lemma:
Let X be compact and A, B disjoint closed subsets of X. Then there exists a continuous functionf :XÑ r0,1swithfpAq t0uandfpBq t1u.
A.5. Lebesgue’s covering lemma: IfpX, dqis a compact metric space andαis is a finite open cover ofX, then there exists aδ ¡0 such that every set AX with diameter diampAq δis contained in some element ofα.
A.6. Category: A subsetA of a topological spaceX is called nowhere dense if the closure of A, denoted by A, has empty interior: ˚A H. A is called offirst category in X if A is the union of countably many nowhere dense subsets of X. A is called of second category in X if it is not of first category. Now let X be a compact or a complete metric space. Then Baire’s category theorem states that every non-empty open set is of second category.
(ii) Measure theory
Somewhat less elementary but even more important for ergodic theory is the con- cept of an abstract measure space. We shall use the standard approach to measure- and integration theory and refer to Bauer [1972] and Halmos [1950]. The advanced reader is also directed to Jacobs [l978]. Although we again assume that the reader is familiar with the basic results, we present a list of more or less known definitions and results.
A.7. Measure spaces and null sets:
A triplepX,Σ, µqis ameasure space ifX is a set, Σσ-algebra of subsets ofX and µa measure on Σ, i.e.
µ: ΣÑR Y t8u σ-additive and andµpHq 0.
If µpXq 8 (resp. µpXq 1), X,Σ, µ is called a finite measure space (resp. a probability space); it is called σ-finite, if X
nPNAn with µpAnq 8 for all nPN.
A setNΣ is aµ-null set ifµpNq 0.
Properties, implications, conclusions etc. are valid “µ-almost everywhere” or for
“almost all xPX” if they are valid for allxPXzN where N is some µ-null set.
If no confusion seems possible we sometimes write “. . . is valid for allx” meaning
“. . . is valid for almost allxPX”.
A.8. Equivalent measures:
LetpX,Σ, µqbe aσ-finite measure space andν another measure on Σ. ν is called absolutely continuous with respect toµif every µ-null set isν-null set. ν is equiv- alent to µ iff ν is absolutely continuous with respect to µ and conversely. The measures which are absolutely continuous with respect to µ can be characterized by the Radon-Nikod`ym theorem (see Halmos [1950],§31).
A.9. The measure algebra:
In a measure spacepX,Σ, µqtheµ-null sets form aσ-idealN . The Boolean algebra Σ :q Σ{N
is called the corresponding measure algebra. We remark that Σ is isomorphic toq the algebra of characteristic functions inL8pX,Σ, µq(see App.B.20) and therefore is acomplete Boolean algebra.
For two subsetsA, B ofX,
A4B : pAYBqzpAXBq pAzBq Y pBzAq denotes the symmetric difference ofA andB, and
dpA, Bq:µpA4Bq
defines a semi-metric on X vanishing on Σ the elements of N (if µpXq 8).
Therefore we obtain a metric onΣ still denoted byq d.
A.10 Proposition: The measure algebrap qZ, dqof a finite measure spaceX,Σ, µ is a complete metric space.
Proof. It suffices to show thatpΣ, dqis complete. For a Cauchy sequencepAnqnPN Σ, choose a subsequence pAniqiPN such that dpAk, Alq 2i for k, l ¡ ni. Then A:8
m1
8
jmAnj is the limit ofpAnq. Indeed, withBm:8
jmAnj we have dpBm, Anmq ¤ ¸8
jm
µpAnj 1zAnjq ¤ ¸8
jm
2j22m
and
dpA, Bmq ¤ ¸8
jm
µpBjzBj 1q ¤ ¸8
jm
dpBj, Anjq dpAnj, Anj 1q dpAnj 1, Bj 1q
¤ ¸8
jm
22j 2j 22pj 1q
¤82m.
Therefore
dpA, Akq ¤dpA, Bmq dpBm, Anmq dpAnm, Akq ¤112m fork¥nm.
A.11. For a subsetW| ofΣ we denote byq ap|WqtheBoolean algebra generated by W|, byσp|WqtheBooleanσ-algebra generated byW|.
Σ is called countably generated, if there exists a countable subsetq W| qΣ such that σp|Wq qΣ.
The metric d relates ap|Wq and σp|Wq. More precisely, using an argument as in (A.10) one can prove that in a finite measure space
σp|Wq ap|Wqd for everyW| qΣ .
A.12. The Borel algebra:
In many applications a set X bears a topological structure and a measure space structure simultaneously. In particular, if X is a compact space, we always take the σ-algebra B generated by the open sets, called theBorel algebra on X. The elements ofBare called Borel sets, and a measure defined onBis a Borel measure.
Further, we only consider regular Borel measures: here, µ is called regular if for everyAPB andε¡0 there is a compact setKAand an open setU Asuch thatµpAzKq εandµpUzAq ε.
A.13 Example:
LetX r0,1sr be endowed with the usual topology. Then the Borel algebraB is generated by the set of all dyadic intervals
D: rk2i,pk 1q 2is:iPN;k0, . . . ,2i1( .
D is called aseparating base because it generates B and for any x, yPX,xy, there isDPD such thatxPD andyRD, orxRD andyPD.
A.14. Measurable mappings:
Consider two measure spaces pX,Σ, µq and pY, T, νq. A mapping ϕ : X Ñ Y is calledmeasurable, ifϕ1pAq PΣ for everyAPT, and calledmeasure-preserving, if, in addition,µpϕ1pAqq νpAqfor allAPTfor allAPT(abbreviated: µϕ1ν).
For real-valued measurable functionsf andgonpX,Σ, µq, whereRis endowed with the Borel algebra, we use the following notation:
rf PBs:f1pBq forBPB, rf gs txPX :fpxq gpxqu, rf ¤gs: txPX:fpxq ¤gpxqu.
Finally,
1A:xÞÑ
#
1 ifxPA
0 ifxRA denotes the characteristic function ofAX. IfAX, we often write1instead of1X. A.15. Continuous vs. measurable functions:
Let X be compact, B the Borel algebra on X and µ a regular Borel measure.
Clearly, every continuous function f :X ÑCis measurable for the corresponding Borel algebras. On the other hand there is a partial converse:
Theorem (Lusin): Letf :XÑCbe measurable andε¡0. Then there exists a compact setAX such thatµpXzAq εandf is continuous onA.
Proof (Feldman [1981]): LettUjujPNbe a countable base of open subsets ofC. Let Vj be open such that f1pUjq V j and µpV Vjzf1pUjqq ε22j. If we take B : 8
j1pVjzf1pUjqq, we obtain µpBq 2ε, and we show that g : f|Bc is continuous. To this end observe that
VjXBcVjX pVjzf1pUjqqcXBcVjX pVjcYf1pUjqq XBc VjXf1pUjq XBcf1pUjq XBc g1pUjq. Since any open subsetU ofCcan be written as U
jPMUj, we haveG1pUq
jPMg1pUjq
jPMVjXBc, which is open in Bc. Now we choose a compact set ABc withµpBczAq ε2, and conclude thatf is continuous onA and that µpXzAq µpBq µpBczAq ε.
A.16. Convergence of integrable functions:
Let pX,Σ, µq be a finite measure space and 1 ¤ p 8. A measurable (real) functionf onX is calledp-integrable, if³
|f|pdµ 8(see Bauer [1972], 2.6.3).
For sequences pfnqnPN of p-integrable functions we have three important types of convergence:
1. pfnqnPNconverges tof µ-almost everywhere if
nlimÑ8pfnpxq fpxqq 0 for almost allxPX.
2. pfnqnPNconverges tof in thep-norm if lim
nÑ8
»
|fnf|pdµ0 see (B.20).
3. pfnqnPNconverges tof µ-stochastically if
nlimÑ8µr|fnf| ¥εs 0 for every ε¡0.
Proposition: LetpfnqnPNbe p-integrable functions andf be measurable.
(i) Iffn Ñf µ-almost everywhere or in thep-norm, thenfnÑf µ-stochastically (see Bauer [1972], 2.11.3 and 2.11.4).
(ii) IfpfnqnPNconverges tof in thep-norm, then there exists a subsequencepfnkq converging tof µ-a.e. (see Bauer [1972], 2.7.5).
(iii) IfpfnqnPN converges tof µ-a.e. and if there is a p-integrable functiong such that |fnpxq| ¤gpxq µ-a.e., then fn Ñf in the p-norm and f is p-integrable (Lebesgue’s dominated convergence theorem, see Bauer [1972], 2.7.4).
Simple examples show that in general no other implications are valid.
A.17. Product spaces:
Given a countable familypXα,Σα, µαqαPAof probability spaces, we can consider the cartesian product X ±
αPA and the so-called product σ-algebra ΣÂ
αPAΣα
which is generated by the set of allmeasurable rectangles, i.e. sets of the form Rα1,...,αnpAα1, . . . , Aαnq: x pxαqαPA:xαiPAαi fori1, . . . , n( forα1, . . . , αnPA, nPN,AαiPΣαi.
The well known extension theorem of Hahn-Kolmogorov implies that there exists a unique probability measureµ:Â
αPAµα on Σ such that µpRα1,...,αnpAα1, . . . , Aαnqq
¹n i1
µαipAαiq
for every measurable rectangle (see Halmos [1950],§383 Theorem B).
ThenX,Σ, µis called theproduct (measure) space defined bypXα,Σα, µαqαPA
Finally, we mention an extension theorem dealing with a different situation (see also Ash [1972], Theorem 5.11.2).
Theorem: Let pXnqnPZ be a sequence of compact spaces, Bn the Borel algebra onXn. Further, we denote by Σ the productσ-algebra onX ±
nPZXn, byFm
the set of all measurable sets inX whose elements depend only on the coordinates m, . . . ,0, . . . , m. Finally we put F
mPNFm. Ifµ is a function on F such that it is a regular probability measure onFmfor eachmPN, thenµhas a unique extension to a probability measure on Σ.
Remark: Let ϕn : X Ñ Yn : ±n
nXi; pxjqjPZ ÞÑ pxn, . . . , xnq. Then we assume above that νnpAq : µpϕn1pAqq, A measurable in Yn, defines a regular Borel probability measure onYn for everynPN.
Proof. The set function µ has to be extended from F to σpFq Σ. By the classical Carath`eodory extension theorem (see Bauer [1972], 1.5) it suffices to show that limiÑ8µpCiq 0 for any decreasing sequence pCiqiPN of sets in F satisfying
iPNCi H. Assume thatµpCiq ¥ε for all iPN and some ε¡0. For eachCi there is ann PN such thatCi P Fn and Ai Yn with Ci ϕn1pAiq. LetBi a closed subset ofAi such thatνnpAizBiq ¤ε22i. ThenDi:ϕn1pBiqis compact in X and µpCizDiq ¤ ε2 2i. Now the sets Gk : k
i1Di form a decreasing sequence of compact subsets ofX, and we have
GkCk andµpGkq µpCkq µpCkzGkq µpCkq µ ¤k
i1
pCizDiq
¥µpCkq
¸k i1
µpCizDiq ¥εε 2 ε
2. Hence Gk H and therefore
iPNCi, which contains
iPNGi, is non-empty, a contradiction.
