Celebratio Mathematica

Joseph L. Doob

Book review: Measure theory by J. L. Doob

by P. A. Meyer

This, the third book by J. L. Doob, fol­lows two pre­vi­ous works. The first, Stochast­ic pro­cesses, had a ma­jor in­flu­ence on twen­ti­eth cen­tury prob­ab­il­ity. The second, Po­ten­tial the­ory and its prob­ab­il­ist­ic coun­ter­part, was the ex­pos­i­tion of two sep­ar­ate but re­lated the­or­ies, to which Doob him­self made cru­cial con­tri­bu­tions. What about this one? It is simply called Meas­ure the­ory and is rather thin (200 pages) but not strik­ingly so (it can­not com­pare with the fam­ous — and long out of print — first edi­tion of Bauer’s Maßthe­or­ie as a Sammlung Göschen, which could be kept in the in­side pock­et of one’s jack­et with the cred­it cards). Its most vis­ible fea­ture upon open­ing it is the re­mark that it is not type­set with \( \mathrm{\TeX} \), with some loss of ty­po­graph­ic­al qual­ity but with the re­fresh­ing im­pres­sion of find­ing an apple or a to­mato on the mar­ket that does not look ex­actly like all apples or to­ma­toes. Then upon read­ing the in­tro­duc­tion we dis­cov­er that “this book was planned ori­gin­ally not as a work to be pub­lished but as an ex­cuse to buy a com­puter…,” and then we may won­der, “If I bought a com­puter, would I write a book on meas­ure the­ory?” The im­plic­a­tion is that the au­thor likes meas­ure the­ory — an eld­erly lady that once had a no­tori­ous af­fair with N. Bourbaki but later came back to a settled life. A book writ­ten by someone that en­joys the sub­ject is likely to be at­tract­ive. Many stu­dents are afraid of meas­ure the­ory; in fact, it is of­ten a turn­ing point in their math­em­at­ic­al edu­ca­tion, as a rather aus­tere con­cen­trate of ep­si­lons and etas. If one really finds it bor­ing, then it may be a good idea to turn to some kind of “geo­metry”.

An old lady in­deed! She was already ma­ture in the thirties, when one got rid of the par­tic­u­lar fea­tures of in­ter­vals and cubes: the second edi­tion of S. Saks’s The­ory of the in­teg­ral (1937) does not dif­fer sub­stan­tially from the books I per­used to write this re­view. All re­cent books on meas­ure the­ory are “good” books, well planned for stu­dents, and they are very much alike. I must say our lib­rary has stopped buy­ing them. It will buy this one. Why?

If we look again at the in­tro­duc­tion, three claims to ori­gin­al­ity are stated. The first one is “the use of pseudo­met­ric spaces”. The book claims to use true func­tions every­where, in­stead of classes of func­tions, which are too of­ten handled slop­pily. It is true that we usu­ally think in this way, and it is good to see someone say­ing it frankly in­stead of the usu­al ex­cuses about us­ing the lan­guage of func­tions while deal­ing with classes. But still classes must be swal­lowed (and you will find them on page 115) in or­der to turn \( L^2 \) in­to an hon­est Hil­bert space. Thus the dif­fer­ence is more psy­cho­lo­gic­al than math­em­at­ic­al.

An­oth­er claim to ori­gin­al­ity is the im­port­ance giv­en to con­ver­gence of meas­ures. It is very good that the au­thor in­cluded the Vi­tali–Hahn–Saks the­or­em and stud­ied weak con­ver­gence of meas­ures on met­ric com­pact and loc­ally com­pact spaces. The more re­cent (Prok­horov) the­ory of tight­ness and weak con­ver­gence on met­ric spaces is not in­cluded — pos­sibly a reas­on­able choice for an ele­ment­ary book.

There are sev­er­al levels of meas­ure the­ory (which in his­tory were suc­cess­ively branded as “empty ab­strac­tion”). The first level is Le­besgue in­teg­ra­tion on the line and on eu­c­lidean space. The second one is the stand­ard gen­er­al meas­ure the­ory, either in its ab­stract form or in its to­po­lo­gic­al form (Radon meas­ures). Doob’s book stops at this level or, more pre­cisely, in­cludes meas­ure the­ory on com­pact met­ric spaces but not on Pol­ish spaces. The third level re­quires ana­lyt­ic sets and ca­pa­cit­ies and is still con­sidered hard math­em­at­ics — which it really is not; they are not even men­tioned in the book, though Doob him­self was among the first to use ca­pa­cit­ies in prob­ab­il­ity the­ory. The fourth level (still un­der act­ive re­search) in­volves a lot of de­script­ive set the­ory and pos­sibly re­fined ax­ioms like the con­tinuum ax­iom or Mar­tin’s ax­iom or the ex­ist­ence of large car­din­als. Again Doob’s choice seems very reas­on­able.

Let me now dis­cuss the main ori­gin­al fea­ture of this book with which I most heart­ily agree. As the in­tro­duc­tion says, “Prob­ab­il­ity con­cepts are in­tro­duced in their ap­pro­pri­ate place, not con­signed to a ghetto.” That is, a stu­dent us­ing this book (after all, it be­longs to the Spring­er col­lec­tion of “Gradu­ate Texts” and is meant to be used by stu­dents) will learn prob­ab­il­ity without even no­ti­cing it, as M. Jourdain was un­wit­tingly talk­ing prose. Most il­lus­tra­tions giv­en in this book are taken from prob­ab­il­ity the­ory, though a few im­port­ant ones (the \( L^2 \) the­ory of Four­i­er series, the Four­i­er–Plancher­el the­or­em, and the ex­ist­ence of ra­di­al lim­its of har­mon­ic func­tions in the disk) are bor­rowed from “pure” ana­lys­is. Even in 1994 say­ing that prob­ab­il­ity the­ory is not only a part of meas­ure the­ory but is the same as meas­ure the­ory is a bold point of view. I think it cor­res­ponds to the truth, but it is worth­while to in­sist a little on the reas­ons why it is not very pop­u­lar.

First of all, prob­ab­il­ity the­ory has deep­er roots in the “real world” than most of math­em­at­ics. His­tor­ic­ally it arose from gambling, which be­longs to the “real” world but is a hu­man activ­ity (“The Lord does not play dice”). Then it in­vaded oth­er forms of hu­man activ­ity (in­sur­ance, eco­nomy, war­fare), then bio­logy (hered­ity), and much more re­cently parts of phys­ics; the prob­lem of quantum mech­an­ics must be left aside, since it in­volves a dif­fer­ent kind of prob­ab­il­ity. Math­em­aticians have a strong tend­ency to be­lieve in the “real­ity” of the ob­jects they are study­ing, wheth­er a tri­angle or a Lie group; oth­er­wise, would they de­vote so much time and thought to them? Gam­blers also have a strong tend­ency to be­lieve in chance. Thus, prob­ab­il­ists have a philo­sophy that dif­fers from that of “pure” math­em­aticians. It is an his­tor­ic­al fact that Kolmogorov’s an­nex­a­tion of prob­ab­il­ity the­ory to meas­ure the­ory was vi­ol­ently re­jec­ted by many prob­ab­il­ists as use­less ab­strac­tion which did not re­spect the ba­sic in­tu­itions of prob­ab­il­ity. When I was a stu­dent, Paul Lévy used to tell us that he could not ac­cept the idea of choos­ing once and for all one single \( \omega \), while it was clear that chance was act­ing all along time. Ac­tu­ally it was Doob, who, be­ing one of the first to take ser­i­ously Kolmogorov’s ap­proach, found the way to re­in­tro­duce time in­to meas­ure-the­or­et­ic lan­guage, as the in­dex set for a fam­ily of \( \sigma \)-fields. This is the sub­ject of the last chapter of the book on mar­tin­gale the­ory.

On the side of non­prob­ab­il­ists, there were sym­met­ric­al ob­jec­tions, not to the math­em­at­ic­al mod­el, of course, (meas­ure the­ory) but to the philo­sophy of prob­ab­il­ity. I found a typ­ic­al ex­ample of this in a beau­ti­ful lec­ture of R. Thom on de­term­in­ism and chaos. After dis­cuss­ing the col­lapse of clas­sic­al de­term­in­ism, he ex­pressed his dis­like of prob­ab­il­ist­ic mod­els as be­ing a dis­hon­est es­cape from the prob­lems of sci­ence. Five minutes later, he was quietly talk­ing about the Li­ouville meas­ure of a hamilto­ni­an sys­tem, and ten minutes later he men­tioned his in­terest in prov­ing struc­tur­al sta­bil­ity in “al­most all” situ­ations. Thus some parts of meas­ure the­ory, or some classes of neg­li­gible sets, seem to be more hon­est than oth­ers.

Even prob­ab­il­ists may have doubts about the “hon­esty” of their do­main. Paul Lévy also used to tell that he did not un­der­stand how an elec­tron could be “free”. The para­dox of prob­ab­il­ity the­ory is that of be­ing a pre­cise math­em­at­ic­al mod­el for something which is un­think­able, namely, an ef­fect without a cause. The elec­tron is not “free”; if it were, there would be a reas­on for its be­ha­viour, while ac­tu­ally there is none. Choos­ing a num­ber at ran­dom means that I have ab­so­lutely no reas­on for choos­ing this num­ber rather than that one, and still I do it.1 After all, there are oth­er ex­amples of math­em­at­ic­al mod­els de­scrib­ing un­think­able pro­cesses — can one “think” with­in our men­tal cat­egor­ies the re­lativ­ist­ic uni­fic­a­tion of space and time or a curved space-time? Still, we handle this in a per­fectly clear and ef­fi­cient sym­bol­ic lan­guage, which is bet­ter ad­ap­ted to the “real world” of phys­ics than the na­ive lan­guage. But ran­dom­ness is also a some­what in­sult­ing idea (“The ref­er­ee wrote he would have ac­cep­ted my pa­per if he had got a double six”), and be­sides that, the math­em­at­ic­al mod­el it­self con­sists of many uni­verses (sample func­tions) of which we see only one. Thus, prob­ab­il­ity the­ory is held in sus­pi­cion by many sci­entif­ic thinkers, who do not ac­cept the claim that some­times “no ex­plan­a­tion” is the best ex­plan­a­tion.

The book com­pletely avoids this kind of philo­soph­ic­al feud. It is strik­ing to see that Doob, one of the great prob­ab­il­ists of our cen­tury, does not claim any priv­ilege for prob­ab­il­ity the­ory with­in meas­ure the­ory. At least, he is care­ful not to give any hint about his own ideas on the sub­ject — maybe he prefers not to think about the un­think­able. At sev­er­al places, he makes a sharp dis­tinc­tion between “math­em­at­ic­al prob­ab­il­ity” and “real prob­ab­il­ity” in terse, iron­ic­al sen­tences. Per­son­ally, I con­sider this very healthy.

On page 80 it is men­tioned that \( n \) drunk­en men are try­ing to re­turn home. The au­thor should have ad­ded or wo­men.

One may hope that, from now on and partly be­cause of this book, it will be im­possible to write a meas­ure the­ory text without in­clud­ing prob­ab­il­ist­ic ideas.

Ed­it­or’s note: P. A. Mey­er (1934–2003) worked at the In­sti­tut de Recher­che Mathématique Avancée (IRMA), at the Uni­versité Louis Pas­teur in Stras­bourg at the time this art­icle was writ­ten.