Von Neumann -On Measure and Ergodic Theory

7/28/2019 Von Neumann -On Measure and Ergodic Theory

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VON NEUM ANN ON MEASURE AND ERGODIC THEORY 8 7Hausdorf f and Banach impl ies tha t the problem of measure has aposi t ive solut ion if n = 1 or n = 2 and a negative solut ion in al l othercases . This f ac t has caused som e m athem at i c i ans to say tha t t hecharac ter of space changes in a fundamenta l and s l ight ly myster iousmanner in the passage f rom two dimensions to three . The purpose ofvon N e um an n ' s m e m oi r [5 ] on the sub jec t is t o pu t the p rob lem in toa proper genera l context and, wi th in tha t context , to analyze and toextend the known (pos i t ive and negat ive) resul t s . The profound ins ight to be ga ined f rom the paper i s tha t the whole problem is es sent ia l ly group- theore t ic in charac ter , and tha t , in par t icular , for thesolvabi l i ty of the problem of measure the ordinary a lgebra ic conceptof so lvabi l i ty of a gro up is re lev ant . Th us , according to von N eu m an n,i t i s the change of group tha t makes a d i f ference , not the change ofspace; replac ing the group of r ig id mot ions by o ther per fec t ly reasonable groups we can produce unsolvable problems in R2 and solvableones in R z.

The r ight way to genera l ize the problem of measure i s to replaceRn by an a rb i t r a ry se t X, to replace the uni t cube (ment ioned in thenorm al i za t ion) by an a rb i t r a ry subse t C of X, and to replace thegroup o f r ig id m ot ions by an a rb i t r a ry g roup G of t r ans fo rm at ionsact ing on X; le t us cal l the general izat ion so obtained the (X, C, G)problem. A useful special case is the (G, G, G) prob lem , w here G isan a rb i t r ar y g rou p cons idered as ac t in g on i tsel f by , say , lef t m ul t ip l i cat ion. If this special problem has a solut ion for a part icular group G,t h e n t h a t g r o u p G i s ca l led "m eas ura ble . " T he genera l problem isredu ced to th e special prob lem in this sen se: if G is a m ea sur ab legroup, then there i s a re la t ive ly s imple condi t ion on X , C , an d G tha tis nec essa ry an d suff icient for th e solvab il i ty of the (X, C, G) problem.(We do not need to s ta te the condi t ion here , but , for purposes ofreference, let us cal l i t the condit ion K.) An abel ian group i s a lwaysm easurab le ; i f N i s a nor ma l subg rou p of G such th a t both N a n dG/N are mea sura ble , the n G is m easu rable . I t fo llows th a t ev erysolvable group i s measurable . Since the condi t ion K t u rns ou t to besat isf ied in the classical cases (r igid motions on Euclidean spaces) ,and s ince the group of r ig id mot ions on Rn i s so lvable exact ly whenn l or w = 2, th e po si t ive solut io n of the classical problem is su bsum ed under the gene ra l i za t ion .

The c lass ica l negat ive resul t s need d i f ferent t rea tment ; the cruc ia lcond i t ion th is t ime i s th e repre sen tabi l i ty in G of a f ree group on twogenera tors . The group of r ig id mot ions on Rn sat isf ies this condit ionexac t ly w hen w >2 , and the un im odula r g roup on R2 also satisfies it.One of the things we can conclude from al l this has a curious sound.


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88 P. R. HALMOSThe rigid motions on R2 are the length-preserving maps and theunimodular transformations are the area-preserving maps, so thatthe length-preserving maps preserve some generalized area whereasthe area-preserving maps do not.If the n egative resu lts are viewed as paradoxical, then there is evena paradox in R1, but it is of a metric (not measure-theoretic) character. Briefly: if E and F are subsets of J?1, write E < F if each of E andF can be decomposed into the same finite number of pieces so thatthere is a one-to-one and distance-decreasing mapping from eachpiece of F onto the corresponding piece of E. Assertion : if E and Fare arbitrary nondegenerate intervals, then E


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VON NEUMA NN ON MEASURE AND ERGODIC THEORY 8 9

they are concrete enough to be applicable to the original (set-theoretic) problem.W e turn now to von N eum ann 's work on analytic, nonpathologicalmeasure theory. The most original item here is the proof of theuniqueness of Haar measure. Compact groups are treated in [14].T he m ethod (later used systema tically in the study of alm ost periodicfunctions) involves the construction of invar iant m eans of continuou sfunctions; the key fact is the compactness of the convex closure of theset of all left translates of each continuous function. The extension ofthe resu lt to locally com pac t groups [16] calls for com pletely differentmethods. The idea now is to replace a given left invariant measure mby a smoothed version m' that has all the assumed properties of m,plus the new and highly desirable property of right zero-invariance.Th e smoothed me asure is defined by m'(E) ~fw(x)rn(Ex)din(x), withan appropriate weight function w; to say tha t it is right zero-invariantmeans that m'{Ex) vanishes exactly when m'{E) does. The possibilityof passing from m to m' is used to reduce the general uniquenessproblem for left invariant measures to the special case of right zero-inv aria nt m easures. After the reduction, the technique is a mod eratelystraightforward application of Fubini's theorem ; the secret of successis that the group, acting on itself by left multiplications, acts er-godically.In a footnote in his 1936 paper on Haar measure von Neumannindicated th at he knew a new proof of the Rad on-N ikodym differentiation theorem, but he did not publish the details of that prooftill four years later. The article [17] is devoted to the construction ofalgebras of operators with peculiar properties. When in the courseof the construction von Neumann found that he needed some measure theory, he cheerfully waded in and started writing a quick textbook on the subject. In six pages we get definitions of all the basicconcepts (e.g., measure and measurable function) and a motivation,statement, and proof of the Radon-Nikodym theorem. Since theproof tidies up a corner of analysis in a completely satisfactory way,and since it deserves to be be tter known, we shall look at the situationin a little more detail.At the center of the stage there are three deservedly celebratedtheorems: the Riesz-Fischer theorem (F ) on the completeness of L2,the Radon-Nikodym theorem (N ) on the differentiability of absolutely continuous set-functions, and the Riesz representationtheorem (R ) for bounded linear functional on L 2 by means of innerproducts. None of these theorems by itself is trivial to prove; the


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90 P. R. HALMOSpleasant and perhaps surprising fact is that the passage from any oneof them to the others is, comparatively speaking, child's play. Thus,in particular, (R ) is a very easy consequence of (N ) ; von Neumann'sproof proceeds in the reverse direction and shows how to derive(N ) from (R).The derivation (for finite positive measures) runs as follows. Suppose that m is absolutely continuous with respect to m', and writern" rn+tn'. If /() =ffdm f then is a bounded linear functional onL%(m") and consequently, by (i?), 1(f) is representable in the formJfgdrn". The Radon-Nikodym derivative of m with respect to m'i s g / ( l - g ) .In his lectures at the Institute for Advanced Study in 1934 vonNeumann discussed measure theory. The mimeographed notes ofthese lectures, despite their limited circulation, were for a long timeone of the major sources of measure-theoretic information in theUnited States. (They became more widely available fifteen yearslater, when they were reproduced and republished in more permanentform [19].) Th ey contain an extremely thorough and detailed exposition of the classical theory of Lebesgue measure in Euclidean spaces,and they contain also the generalization of the theory to abstractmeasure spaces. Except for a few technical details (e.g., the conceptof a semiring) the work is not original. It does, however, containresults that it is difficult to find anywhere else. It contains, for instance, a study of the connection between set-functions and point-functions (i.e., the theory of "cumulative distribution functions") inRn, which is an important and complicated subject that most booksavoid, or, at best, treat in the trivial one-dimensional case only.

In 1940 von Neumann lectured on measure theory again. The titleof the course was Invariant measures. The notes were typed, and atleast one copy is in the library of the Institute for Advanced Study,but they were never published. The main emphasis of the course wason measures in locally compact spaces and groups. Four of the sixchapters are an exposition of the by now well known results of Haarand Weil. The other two chapters digress from the main theme, and,incidentally, they con tain original work. Ch apte r I I tre ats generalizedlimits. The first purpose of this chapter was to serve as an auxiliaryin the proof of the existence of Haar measure, but the treatment ismuch more comprehensive than that single application requires.Chapter VI contains a construction of Haar measure by means of"approximately equi-distributed" finite sets; some of the results ofthis chapter were obtained in collaboration with Kakutani.


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VON NEUM ANN ON MEASURE AND ERGOD1C THEORY 9 1

We com e now to von N eum ann ' s w ork on e rgod ic theory . Thema jor p ar t of th is work was don e in the ear ly 1930s ; wi th o ne except ion a l l h is publ ica t ions on the subjec t appeared in 1932.M od ern ergodic theo ry s tar ted ear ly in 1931 wi th a mo st s ignif icantobse rva t ion m ad e by Ko opm an [3 ] . T he obse rva t ion i s t h a t t hefunc t iona l op e ra to r induced by a m easure -p rese rv ing t r ans fo rm a t ionis un i ta ry . If, in oth er wo rds, T i s a me asure-p reserving t rans form at ion on some measure space , and i f for every square- in tegrable funct ion on th a t space a new squ are- in te grab le funct ion Uf is definedb y (Uf)(x) =f(Tx), t h e n U i s a uni tary opera tor on L 2 . By th is t imevon N eum ann had w r i t t en h i s epoch-m aking a r t i c l e s on ope ra to rtheory and he knew eve ry th ing abou t the sub jec t t ha t w as thenknow n. Koopm an ' s obse rva t ion w as s im ul t aneous ly a cha l l enge anda h in t . If the re i s an in t im ate connect ion be tw een mea sure-pres ervingt rans fo rm at ions and un i t a ry ope ra to r s , t hen the know n ana ly t i ctheory of such opera tors must sure ly g ive some informat ion aboutthe geometr ic behavior of the t ransformat ions . By October of 1931von N eum ann had the answ er ; t he answ er w as the m ean e rgod ict h e o r e m .A s s t a t ed now adays , t he m ean e rgod ic theorem says tha t i f U is au n i t a r y o p e r a t o r o n a H u b e r t s p a c e H, then the sequence of averages

U+U/+ - . . + U"-*f)ni s s t ron gly con ver gen t for every in H. (The mean ergodic theoremalso says something about what the l imi t has to be , and the convergen ce asser t ion can ju s t as well be m ad e for su i tab le on e-pa ram eterfamilies {lit) in place of the sequences { U n}. These ref inements arei r re levant in a summary such as the present one and wi l l be system at ica l ly ignored. ) In von N eu m an n 's f irs t no te [8] on the subjec tthe theorem w as no t s t a t ed in th i s abs t r ac t fo rm ; he em phas ized themeasure- theore t ic roots of the resul t throughout . A reader of thenote could have fo l lowed the ent i re a rgument , involving cont inualreferences to the under ly ing measure space , wi thout ever becomingaw are tha t t he p roof p roved a m ore abs t r ac t t heorem than the s t a t e m e n t s t a t e d .

Shor t ly a f t e r von N eum ann p roved the m ean e rgod ic theorem hediscussed i t with G. D. Birkhoff, and shor t ly af te r tha t B i rkhoffproved the " individual" e rgodic theorem [ l ] . (The order of events ashere s ta ted would be imposs ib le to guess f rom the da tes appear ingon the var ious publ ica t ions ; the pr ior i ty s i tua t ion i s expla ined in a


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92 P. R. HALMOSsu bse qu en t no te [2] of Bi rkhoff an d K oop ma n. ) T h e di fference betwee n von N eu m an n ' s resu l t and B irkhofFs is easy to descr ibe ; theformer s ta tes tha t for su i table funct ions the sequence of averages

- ((*) +f(Tx) + + f(T^x))ni s con verg ent in th e mea n of order 2 , wh ereas the la t te r s ta tes th a tthe same sequence i s convergent for a lmost every x. Thus B i rkhofFsresu l t is i nex t r i cab ly t ied up w i th m e asure ; von N e um an n ' s can be(and repeatedly has been) genera l ized to Banach spaces and otheraxio ma t ic con texts . I t i s therefore cur ious , bu t t rue , th a t von Ne umann a lways looked a t e rgodic theory as a par t of measure theory ; henever w orked on the abs t r ac t ve r s ions . Wha t f a sc ina ted h im m os tw as the de l i ca t e in t e rp lay be tw een m easure and spec t rum . Theergodic theorem i t se l f (mean or individual ) was a lmost never neededin his later work; i ts main role was that of historical just i f icat ion fors tudy ing m easu re -p rese rv ing t r ans fo rm at ion s .To under s t and the na tu re o f von N eum ann ' s p ro found ins igh t sin to the s t ruc ture of measure-preserving t ransformat ions , le t us lookat some of the most s t r ik ing and typica l ones . For s impl ic i ty of s ta tem en t w e re s t r i c t a t t en t ion to m easure spaces of to t a l m easure 1 ; w i tha l i t t le care many of the resul ts can be extended to infini te spaces.Associa ted wi th every measure space there i s a measure a lgebra ,namely the Boolean a lgebra of measurable se ts modulo se ts of measure zero . A measure-preserving t ransformat ion of the space inducesin a na tu ra l w ay a m easure -p rese rv ing au tom orph i sm of the a lgebra .In h ighly pa thologica l measure spaces i t can happen [18 ] t h a t t h espace has few er t r ans fo rm at ion s th an th e a lgebra has a u tom orph i sm s ,i .e . , th a t no t eve ry mea sure-pres erving au tom orp hism is induced by am easure -p rese rv ing t r ans fo rm at ion . I f t he under ly ing m easure spaceis subjec ted to some ra ther reasonable topologica l and metr ic condi t i ons , t hen the pa tho logy canno t happen . V on N eum ann ' s p roof o fth is useful fact [ l l ] is a fr ightful ly com plicate d set- the ore t ic argu ment . The resul t has not ye t been improved in any essent ia l way.The m os t in t e res t ing m easure -p rese rv ing t r ans fo rm at ions a re theergodic ones ; e rgodic i ty i s the na tura l measure- theore t ic genera l izat ion of the concept of t rans i t iv i ty tha t occurs in the theory of per m uta t ion g roups . (To be p rec i se , T is ergodic if and only if the onlym e a s u r a b l e s e t s i n v a r i a n t u n d e r T are the sets of measure zero andthei r com plem ents . ) In h is longes t an d m ost d ivers if ied pap er one rgod ic theo ry [12] von N e um an n show ed tha t on su i t ab ly r e s t r ic t edspaces eve ry m easure -p rese rv ing t r ans fo rm at ion can be decom posed


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VON NEUMANN ON MEASURE AND ERGODIC THEORY 93into ergodic pieces. The theorem is a forerunner of many such decomposition (or "direct integral") theorems.

The mixing theorem (obtained in collaboration with Koopman[9]) is a typical example of the measure-versus-spectrum type ofresult; it asserts that a geometric property of T (mixing) is equivalentto a spectral property of U (no non trivial proper values) . In somewhat informal but quite suggestive language, the mixing theoremsays that a necessary and sufficient condition that each pair of setsbe eventually stochastically independent is that the spectrum beessentially continuous.

There is also a beautiful result about the very opposite situationin which U has pure point spectrum (i.e., L% has an orthonormalbasis consisting of proper vectors of U). For ergodic transformationsof this kind it turns out [12] that the spectrum, which is always asubset of the set of complex numbers of modulus 1, is in fact a subgroup of the multiplicative group of such numbers, and every suchsubgroup is the spectrum of some such transformation. Even more istrue; the Koopman program is really fulfilled here. If both S and Tsatisfy the conditions (ergodic, pure point spectrum) then a necessaryand sufficient condition for the measure-theoretic isomorphism of Sand T is the unitary equivalence of the corresponding unitary operators. In other words, for a special but large class of transformationsthe analytic (operator) methods give complete information about thegeometric (transformation) questions.

This concludes our summary of von Neumann's contributions tomeasure and ergodic theory. In quantity these contributions amountroughly to one tenth of von Neumann's scientific publications. As totheir quality, it seems to be safe to say that if von Neumann hadnever done anything else, they would have been sufficient to guarantee him mathematical immortality.

B I B L I O G R A P H Y[Except for three notes by Birkhoff and Koopman, added for historical reasons,what follows is a complete list of von Neumann's publications on measure theory andon the mathematical aspects of ergodic theory. Not every item in the list is referredto in the text.]1. G. D. Birkhoff, Proo f of the ergodic theorem, Proc. Nat. Acad. Sci. vol. 17 (1931)pp. 656-660.2. G. D. Birkhoff and B. O. Koopman, Recent contributions to the ergodic theory,Proc. N at. Acad. Sci. vol. 18 (1932) pp. 279-282.3. B. O. Koopman, Hamiltonian systems and transformations in Hilbert space,Proc. N at. Acad. Sci. vol. 17 (1931) pp. 315-318.4. John von Neumann, Die Zerlegung eines Intervalles in abzahlbar viele kongruenteTeilmengen, Fund. Math. vol. 11 (1928) pp. 230-238.


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94 P. R. HALMOS5. , Zur allgemeinen Theorie des Masses, Fund. Math. vol. 13 (1929) pp.73-116.6. , Zusatz zur Arbeit "Zur allgemeinen Theorie des Masses," Fund. Math.vol. 13 (1929) p. 333.7. , Algebrasche Reprasentanten der Funktionen (lbis auf eine Menge vomMasse Null,'1 J. Reine Angew. Math. vol. 165 (1931) pp. 109-115.8. , Pro of of the quasi-ergodic hypo thesis, Proc. Nat. Acad. Sci. vol. 18(1932) pp. 70-82.9. and B. O. Koopm an, Dynamical systems of continuous spectra, Proc.Nat. Acad. Sci. vol. 18 (1932) pp. 255-263.10. , Physical applications of the ergodic hypothesis, Proc. Nat. Acad. Sci.vol. 18 (1932) pp. 263-266.11. , Einige Sdtze ber messbare Abbildungen, Ann. of Math. vol. 33 (1932)pp. 574-586.12. , Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math.vol. 33 (1932) pp. 587-642.13. , Zusatze zur Arbeit uZur Operatorenmethode . . . ," Ann. of Math. vol.33 (1932) pp. 789-791.14. , Zum Haarschen Mass in Topo logischen Gruppen, Compositio Math.vol. 1 (1934) pp. 106-114.15# a n c [ M. H. Stone, The determination of rep resentative elements in the

residual classes of a B oolean algebra, Fund. Math. vol. 25 (1935) pp. 353-378.16. , The uniqueness of Haar's measure, Mat. Sbornik vol. 1 (1936) pp.721-734.17# f Q n ringS 0f operators. Ill, Ann. of Math. vol. 41 (1940) pp. 94-161.13# a n ( j p # R # Halmos, Operato r methods in classical mechanics, II , A nn. ofMath. vol. 43 (1942) pp. 332-350.19# 1 Functional operators. I. Measures and integrals, Princeton, 1950.UNIVERSITY OF CHICAGO AND

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