#### 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.*