Émile Borel

How to use this course

Émile Borel’s name is attached to more objects in a first graduate course than almost anyone else’s: Borel sets, Borel measures, the Heine–Borel theorem, the Borel–Cantelli lemma, Borel summation. Each of these began as a short, concrete argument that Borel made and then left for others to generalize. He was also a public figure: an editor, a deputy in the French parliament, Minister of Marine, and the founder of institutions that still shape French science.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 07 follow Borel’s work in analysis (1894–1901); Lessons 08 to 10 follow his turn to probability after 1905.

Timeline

Saint-Affrique to Paris, 1871–1889

Félix Édouard Justin Émile Borel was born on 7 January 1871 in Saint-Affrique, a small town in the Aveyron in southern France. His father, Honoré Borel, was a Protestant pastor who ran a small school for the children of Protestant families in the family home. His mother, Émilie Teissié-Solier, came from a family of wool merchants of the town. He had two sisters, both much older than him.

He learned to read and count in his father’s school. At eleven he went to the lycée at Montauban, where he lived with his eldest sister, Madame Lebeau. His teachers saw quickly that he was exceptional, and he was sent to Paris on a scholarship to the Collège Sainte-Barbe, following courses at the Lycée Louis-le-Grand. Sainte-Barbe prepared boys for the competitive examinations of the grandes écoles.

In 1889 he came first in the entrance examinations of both the École polytechnique and the École normale supérieure, and he won the Concours général in mathematics. The two schools offered different futures: Polytechnique led to engineering and the state corps, the École normale to teaching and research. He chose the École normale. The choice set him on the path of Charles Hermite, Henri Poincaré and his own future teacher Gaston Darboux.

The École normale, the thesis, Lille, 1889–1897

At the École normale Borel came first in the agrégation in 1892 and wrote his doctoral thesis under Darboux, Sur quelques points de la théorie des fonctions, defended in 1893 and published in the Annales de l’École normale in 1895.

The thesis was about analytic continuation. Borel studied series such as \(\sum_n A_n/(z-a_n)\) whose poles \(a_n\) are dense along a curve. Weierstrass and Poincaré had treated such curves as natural boundaries beyond which a function cannot be continued. Borel showed that if the \(A_n\) decrease fast enough, the series still converges at most points of the curve and can define a function that passes through it in a weaker sense. To prove that “most points” survive, he needed two new tools: a covering theorem for intervals and the idea that a countable set has length zero. Both reappear in Lessons 04 and 05. The thesis also contains what is now called Borel’s lemma on smooth functions with prescribed Taylor series (Lesson 06).

In 1893 he became maître de conférences at the University of Lille. In three or four years there he published 22 papers. Among them was his 1896 proof of Émile Picard’s theorem that a non-constant entire function omits at most one complex value. Picard’s proof of 1879 used the elliptic modular function. Borel found a proof by elementary estimates of growth, which had been sought for almost two decades and which opened the theory of value distribution later completed by Rolf Nevanlinna.

Paris, the Appells and the Revue du mois, 1897–1914

In January 1897 Borel returned to the École normale as maître de conférences. His lectures became books: the Leçons sur la théorie des fonctions (1898) set out his theory of measure, and Leçons sur les séries divergentes (1901) his summation methods. These were the first volumes of the “Borel collection” of monographs on the theory of functions, which ran to dozens of volumes by many authors, including Henri Lebesgue, René Baire and Paul Montel. The Académie des sciences gave him its Grand Prix in 1898, the Poncelet Prize in 1901, the Vaillant Prize in 1904 and the Petit d’Ormoy Prize in 1905.

In October 1901 he married Marguerite Appell, the seventeen-year-old daughter of the mathematician Paul Appell. Under the pen name Camille Marbo, built from the first letters of Marguerite and of Borel, she wrote more than thirty novels and won the Prix Femina in 1913 for La Statue voilée. The couple had no children of their own. In 1906 they used the money from one of his prizes to found La Revue du mois, a monthly review of science, letters and politics that ran until 1920.

From 1905 Borel turned to probability. His 1909 paper on denumerable probabilities (Lesson 08) was the first to apply measure theory to infinite sequences of random trials. The same year the Sorbonne created a chair of the theory of functions for him, and in 1910, on the death of Jules Tannery, he became deputy director of the École normale, responsible for its scientific students. He also became interested in the new physics: his book Introduction géométrique à quelques théories physiques (1914) used hyperbolic geometry to explain the composition of velocities in special relativity.

The covering theorem

For the general reader. If you try to cover a closed stretch of the number line with infinitely many small open windows, you never actually need infinitely many. Some finite collection already does the job. Borel proved this for countably many windows in his thesis. The result is the seed of the modern notion of compactness.

Proof sketch

Let \(S\) be the set of \(x\in[a,b]\) such that \([a,x]\) is covered by finitely many \(I_n\). Then \(a\in S\), and \(S\) is an interval starting at \(a\). Let \(c=\sup S\). Some \(I_m\) contains \(c\), say \(I_m=(\alpha,\beta)\) with \(\alpha<c<\beta\). Choose \(x\in S\) with \(x>\alpha\); the finite cover of \([a,x]\) together with \(I_m\) covers \([a,y]\) for every \(y<\beta\). Hence \(c\in S\), and if \(c<b\) then points beyond \(c\) are in \(S\), a contradiction. So \(c=b\in S\).

The argument uses only the least upper bound property of \(\mathbb R\), and it works equally well for an arbitrary family of open intervals. Pierre Cousin (1895) and Lebesgue (1904) gave such general versions. Eduard Heine had used a related argument in 1872 to prove that a continuous function on a closed interval is uniformly continuous, and the result became known as the Heine–Borel theorem. In \(\mathbb R^n\) it says that closed bounded sets are compact: every open cover has a finite subcover.

Borel’s use of the theorem

By the theorem, finitely many \(I_1,\dots,I_N\) cover \([a,b]\), and for finitely many intervals the inequality \(\sum_{n\le N}\ell_n\ge b-a\) follows by induction on \(N\). Now enumerate the rationals \(r_1,r_2,\dots\) of \([0,1]\) and put an open interval of length \(\varepsilon 2^{-n}\) around \(r_n\). These cover all rationals and have total length at most \(\varepsilon\). For \(\varepsilon<1\) the corollary says they miss points of \([0,1]\). This is the picture at the top of the page, and it is the first appearance of a set of measure zero.

Measure and Borel sets

For the general reader. How long is a set of points that is not an interval? Borel proposed simple rules: a countable union of non-overlapping pieces has the total length of the pieces, and removing a piece subtracts its length. The sets reachable from intervals by these rules are now called Borel sets. Lebesgue built his integral on this idea four years later.

Borel’s programme (1898)

In the Leçons sur la théorie des fonctions Borel asked for a measure \(m\) on sets of real numbers with three properties: the measure of an interval is its length; if \(E_1,E_2,\dots\) are pairwise disjoint, \(m(\bigcup E_n)=\sum m(E_n)\); and if \(F\subseteq E\), then \(m(E\setminus F)=m(E)-m(F)\). He called sets obtainable from intervals by repeated countable disjoint unions and differences “measurable”. Countable additivity, the second rule, was the decisive step. Camille Jordan’s earlier content was only finitely additive, and the rationals in \([0,1]\) had no Jordan content at all.

Borel did not prove that his rules were consistent, that is, that no set receives two different measures. Lebesgue settled this in his 1902 thesis, written under Borel’s guidance, by defining outer measure through countable covers by intervals, exactly the device of Lesson 04.

Existence follows from Lebesgue’s outer measure together with Carathéodory’s criterion (1914); uniqueness follows because intervals form a class closed under intersection that generates \(\mathcal B(\mathbb R)\). Countable sets have measure 0 by Borel’s covering argument: a point is an interval of length 0, and countable additivity does the rest.

The Borel hierarchy

Open sets are countable unions of open intervals. Countable intersections of open sets are called \(G_\delta\), countable unions of closed sets \(F_\sigma\), and the process continues transfinitely through all countable ordinals. Every level is contained in \(\mathcal B(\mathbb R)\), and \(\mathcal B(\mathbb R)\) has the cardinality of the continuum. Since the Cantor set has measure 0 and continuum many points, it has \(2^{\mathfrak c}\) subsets, all Lebesgue-null. So most Lebesgue-measurable sets are not Borel. Descriptive set theory, founded by Baire, Lebesgue, Nikolai Luzin and Mikhail Suslin, studies this hierarchy.

Power series and entire functions

For the general reader. A Taylor series records all the derivatives of a function at one point. For analytic functions the series determines the function. Borel showed that for merely smooth functions anything goes: every sequence of numbers is the list of derivatives of some smooth function. He also found elementary ways to control how fast entire functions grow, and used them to give a new proof of Picard’s theorem.

Proof sketch

Fix \(\chi\in C^\infty(\mathbb R)\) with \(\chi=1\) on \([-\tfrac12,\tfrac12]\) and \(\chi=0\) outside \([-1,1]\). Put

with \(\lambda_n\ge1\) growing fast. Each term \(g_n(x)=\frac{c_n}{n!}x^n\chi(\lambda_nx)\) agrees with \(\frac{c_n}{n!}x^n\) near 0, so it contributes \(c_n\) to the \(n\)-th derivative at 0 and nothing to the others. The term is supported in \(|x|\le1/\lambda_n\), and for \(k<n\) one checks \(\sup|g_n^{(k)}|\le C_{n,k}|c_n|\lambda_n^{k-n}\), where \(C_{n,k}\) depends only on \(\chi\), \(n\) and \(k\). Choosing \(\lambda_n\) so that \(\sup|g_n^{(k)}|\le2^{-n}\) for all \(k<n\) makes every differentiated series converge uniformly, so \(f\) is smooth and term-by-term differentiation is justified.

Growth of analytic functions

The inequality bounds a function using only an upper bound on its real part. It carries the names of Borel and Constantin Carathéodory, who gave the sharp form. For the proof, assume \(f(0)=0\) and \(A=A(R)>0\). Then \(\phi(z)=f(z)/(2A-f(z))\) maps the disc \(|z|<R\) into the unit disc with \(\phi(0)=0\), so the Schwarz lemma gives \(|\phi(z)|\le r/R\) on \(|z|=r\); solving for \(f\) gives \(|f|\le 2Ar/(R-r)\). The general case follows by applying this to \(f-f(0)\).

Picard’s proof composed \(f\) with the inverse of the elliptic modular function and applied Liouville’s theorem. Borel’s 1896 proof avoided this and worked by comparing growth rates. If \(f\) omits 0 and 1, then \(f=e^{g}\) and \(1-f=e^{h}\) for entire \(g,h\), so \(e^{g}+e^{h}=1\). Borel then proved general theorems showing that identities among exponentials of entire functions force the functions to be constant. These ideas led to Nevanlinna’s theory of value distribution in the 1920s.

Summing divergent series

For the general reader. The series \(1-1+1-1+\cdots\) has no sum in the usual sense, yet physicists and astronomers kept meeting divergent series that gave excellent numerical answers. Borel found a method that assigns sums to many such series by turning them into integrals. His method is still used in quantum field theory, where perturbation series typically diverge.

The method (1899)

The identity \(\int_0^\infty t^n e^{-t}\,dt=n!\) suggests writing \(a_n=\int_0^\infty \frac{a_n t^n}{n!}e^{-t}\,dt\) and exchanging sum and integral.

The method is regular: if \(\sum a_n\) converges to \(s\), its Borel sum exists and equals \(s\). It is also linear. Borel developed the method in his 1899 memoir and in his Leçons sur les séries divergentes (1901). He was not the first to assign sums to divergent series. Euler had done so freely, and Ernesto Cesàro had introduced averaging methods. Borel’s contribution was a method strong enough to sum series with factorially growing terms.

Power series and the Borel polygon

Apply the method to a power series \(\sum a_nz^n\) with sum \(f(z)\) near 0. For each singular point \(a\) of \(f\), draw the line through \(a\) perpendicular to the segment \([0,a]\), and keep the open half-plane containing 0. The intersection of these half-planes is the Borel polygon of \(f\).

The polygon contains the disc of convergence and is usually much larger. For uniqueness, George Neville Watson (1912) and Alan Sokal (1980) showed that a function analytic and suitably bounded in a sector is determined by its asymptotic series, so the Borel sum is the only “natural” value. In modern resurgence theory, singularities of \(\mathcal B\) on \([0,\infty)\) obstruct summation and encode nonperturbative effects.

Denumerable probabilities and normal numbers

For the general reader. Toss a fair coin forever. Will the proportion of heads tend to one half? Borel showed in 1909 that it does with probability one. Translated into numbers, almost every real number has digits in which every pattern appears with exactly the expected frequency. Yet it is very hard to prove this for any particular number such as \(\pi\).

The model

In Les probabilités dénombrables et leurs applications arithmétiques (1909) Borel considered countably many trials, where earlier probability had dealt with finitely many. Write \(x\in[0,1]\) in binary, \(x=\sum_k d_k(x)2^{-k}\). Under Lebesgue measure the digits \(d_1,d_2,\dots\) are independent and each equals 1 with probability \(\tfrac12\). So an infinite coin-tossing sequence is a random point of \([0,1]\), and probabilities of events about the whole sequence are measures of sets.

Part (i): \(P(\limsup A_n)\le P(\bigcup_{n\ge N}A_n)\le\sum_{n\ge N}P(A_n)\to0\). Part (ii): \(P(\bigcap_{n\ge N}A_n^c)=\prod_{n\ge N}(1-P(A_n))\le\exp(-\sum_{n\ge N}P(A_n))=0\). Borel worked with independent events; Francesco Paolo Cantelli stated part (i) in general in 1917.

Proof by the fourth moment

Let \(X_k=d_k-\tfrac12\), independent with mean 0, \(E X_k^2=\tfrac14\) and \(E X_k^4=\tfrac1{16}\). Expanding \((X_1+\cdots+X_n)^4\), all terms containing some \(X_k\) to the first power have mean 0, so

By Markov’s inequality, \(P(|S_n/n-\tfrac12|>\varepsilon)\le\frac{3}{16\,\varepsilon^4n^2}\), which is summable. By Borel–Cantelli (i), almost surely \(|S_n/n-\tfrac12|>\varepsilon\) only finitely often. Take \(\varepsilon=1/m\) for \(m=1,2,\dots\) and intersect the countably many full-measure sets. Borel’s own argument had gaps. Georg Faber (1910) and Felix Hausdorff (1914) gave rigorous proofs, and the fourth-moment argument above is the standard modern version.

Normal numbers

A number \(x\) is normal in base \(b\) if every block of \(k\) digits occurs in its base-\(b\) expansion with limiting frequency \(b^{-k}\), and absolutely normal if it is normal in every base \(b\ge2\).

For a fixed base and block, the same argument as above applies to the indicator of the block at positions \(k\) apart, giving full measure. There are countably many bases and blocks, and a countable intersection of full-measure sets has full measure. Explicit examples are harder. David Champernowne showed in 1933 that \(0.123456789101112\ldots\) is normal in base 10. Whether \(\pi\), \(e\) or \(\sqrt2\) is normal in any base is unknown.

Games of strategy

For the general reader. In games like poker, the outcome depends on both chance and the players’ choices. Between 1921 and 1927 Borel wrote a series of notes in which he defined games of strategy and suggested that the best play is often to randomize, choosing among moves with fixed probabilities. He solved small cases and asked whether a best mixed strategy always exists. John von Neumann proved in 1928 that it does.

Borel’s formulation

In his 1921 note in the Comptes rendus, Borel considered two players who each choose one of \(n\) “methods of play”. If player I chooses \(i\) and player II chooses \(j\), player I wins \(a_{ij}\) and player II loses it. Borel studied symmetric games, where the roles can be exchanged, so the payoff matrix \(A=(a_{ij})\) is skew-symmetric: \(a_{ji}=-a_{ij}\). A mixed strategy is a probability vector \(x\in\Delta_n=\{x\ge0,\ \sum x_i=1\}\). If the players use \(x\) and \(y\), the expected gain of player I is \(x^{\mathsf T}Ay\).

Borel showed that for \(n=3\) and, in a later note, for \(n=5\), each player has a mixed strategy guaranteeing an expected gain of at least 0. At first he doubted that this extended to all \(n\). By 1927 he regarded the general question as open, having found no counterexample.

Proof sketch

The inequality \(\max\min\le\min\max\) is elementary. The key step is an alternative: either some \(y\in\Delta_n\) has \(Ay\le0\) in every coordinate, so \(\min_y\max_x x^{\mathsf T}Ay\le0\), or some \(x\in\Delta_m\) has \(x^{\mathsf T}Ay>0\) for all \(y\), so \(\max_x\min_y x^{\mathsf T}Ay>0\). To prove it, let \(K=\{Ay+u : y\in\Delta_n,\ u\ge0\}\), a closed convex subset of \(\mathbb R^m\). If \(0\in K\), the first case holds. If not, a hyperplane separates 0 from \(K\): there is \(z\) with \(z\cdot k>0\) for all \(k\in K\). Letting \(u\) grow along each coordinate shows \(z\ge0\), and normalizing gives \(x\in\Delta_m\) in the second case. Applying the alternative to \(A-cJ\), where \(J\) is the all-ones matrix, for every real \(c\) shows that no \(c\) can lie strictly between \(\max\min\) and \(\min\max\). For skew-symmetric \(A\), \(x^{\mathsf T}Ax=0\) for all \(x\), so \(v\le0\le v\).

The priority question became public in 1953, when Econometrica published English translations of three of Borel’s notes with commentary by Maurice Fréchet, followed by a reply from von Neumann. Borel had the concepts of pure and mixed strategy; von Neumann had the theorem. In 1938 Borel also analysed a simplified poker game, “la relance”, in his book on applications of probability to games of chance.

Kinetic theory, paradoxes and monkeys

For the general reader. Borel thought hard about what probability means in physics and in daily life. He showed how the bell-shaped distribution of molecular speeds follows from pure geometry in many dimensions. He pointed out a puzzle about conditioning on events of probability zero. And he invented the image of monkeys at typewriters to explain why some possible events never happen in practice.

Gas molecules and high-dimensional spheres

In the kinetic theory of gases, \(N\) particles of fixed total kinetic energy have a velocity vector on a sphere in \(\mathbb R^{3N}\). Borel argued in 1906 that a uniformly distributed point on such a sphere has nearly Gaussian coordinates, recovering Maxwell’s distribution from geometry. The result is often called the Poincaré–Borel lemma.

The slice of the sphere at height \(x\) is a sphere of dimension \(n-2\) and radius \(\sqrt{n-x^2}\), and the surface element contributes a factor \(\sqrt n/\sqrt{n-x^2}\). This gives the density \((n-x^2)^{(n-3)/2}\) up to a constant. Then \((1-x^2/n)^{(n-3)/2}\to e^{-x^2/2}\).

The Borel–Kolmogorov paradox

In his 1909 textbook Éléments de la théorie des probabilités, Borel asked: choose a point uniformly on the Earth’s surface and suppose it lies on a given great circle. What is its conditional distribution on that circle? Symmetry suggests uniform. But if the circle is a meridian, conditioning by longitude gives a different answer. Andrey Kolmogorov explained in 1933 that conditioning on an event of probability zero is defined only relative to a family of events, a \(\sigma\)-algebra, and different families give different answers.

The typing monkeys

In a 1913 paper on statistical mechanics and irreversibility, Borel imagined a million monkeys typing ten hours a day at random. They might in principle reproduce the books of the world’s great libraries, yet the probability is so small that the event can be treated as impossible. He used the image to explain why a gas never spontaneously gathers in one half of its container, although the laws of mechanics allow it. Late in life, in Les probabilités et la vie (1943), he raised this into a principle: sufficiently small probabilities should be treated as zero at the human, terrestrial and cosmic scales.

War, politics and institutions, 1914–1940

When war broke out in 1914 Borel was 43. He volunteered, commanded an artillery battery, and organized sound-ranging, the location of enemy guns by the arrival times of their reports at several microphones. In 1915 his friend Paul Painlevé, a mathematician and now a minister, put him at the head of the technical office of a new directorate of inventions for national defence. When Painlevé’s government fell in 1917, Borel went back to the front. His nephew and adopted son Fernand Lebeau was killed in 1915. Borel received the Croix de guerre in 1918.

After the war he changed direction. In 1920 he resigned as deputy director of the École normale. In 1921 he moved to the Sorbonne chair of probability and mathematical physics, succeeding Joseph Boussinesq, and was elected to the Académie des sciences. In 1922 he founded the Institut de statistique de l’Université de Paris, the first French school of statistics, and from 1924 he edited a multi-volume Traité du calcul des probabilités et de ses applications.

He also entered politics. He was elected to the Chamber of Deputies for the Aveyron in 1924, on the republican centre-left, and sat until 1936. He served as mayor of Saint-Affrique. From 17 April to 28 November 1925 he was Minister of Marine in Painlevé’s government. As a deputy he argued for state funding of research, and his efforts, with those of the physicist Jean Perrin, prepared the ground for the national research funds of the 1930s and the founding of the CNRS in 1939.

His most lasting institution was the Institut Henri Poincaré, opened in Paris in 1928 with money from the Rockefeller Foundation and the Rothschild family. It was devoted to probability and mathematical physics, and Borel led it from its opening. In the 1930s it became the home of the French school of probability, including Maurice Fréchet and the young Wolfgang Doeblin. Borel was president of the Académie des sciences in 1934.

Occupation, last years and legacy, 1940–1956

Borel left his Sorbonne chair in 1941. That year, under the occupation, he was arrested and held at Fresnes prison for about a month. On his release he returned to Saint-Affrique and worked with the Resistance in the Aveyron. After the Liberation he received the Médaille de la Résistance.

In his seventies he wrote a stream of books for a wide audience: Le jeu, la chance et les théories scientifiques contemporaines (1941), Les probabilités et la vie (1943), Les paradoxes de l’infini (1946) and Les nombres inaccessibles (1952). The last argued that most real numbers can never be individually named. He was elected to the Bureau des longitudes in 1946, chaired the science committee of UNESCO from 1948, and received the Grand Cross of the Légion d’honneur in 1950. In 1955 the CNRS awarded him its first gold medal.

In 1955 he travelled to Brazil to present a paper at a session of the International Statistical Institute. On the voyage home he fell on board ship. He never fully recovered and died in Paris on 3 February 1956, aged 85. He is buried at Saint-Affrique.

Legacy

• Measure and integration: countable additivity, Borel sets and sets of measure zero made Lebesgue’s integral possible and are the language of modern analysis.
• Topology: the covering theorem became the definition of compactness.
• Probability: the 1909 paper began probability as measure theory, the framework Kolmogorov axiomatized in 1933; Borel–Cantelli is in every probability course.
• Divergent series: Borel summation is a standard tool in asymptotics, resurgence and quantum field theory.
• Game theory: pure and mixed strategies, defined in his notes of 1921–1927.
• Students: Henri Lebesgue wrote his thesis under Borel’s guidance; Paul Montel, Georges Valiron, Georges Bouligand, Ernest Esclangon, Francis Perrin and Tadeusz Ważewski were among his doctoral students.
• Institutions: the Institut Henri Poincaré, the Paris institute of statistics (now ISUP), and a role in the creation of the CNRS. A lunar crater bears his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Émile Borel, MacTutor History of Mathematics Archive.
2. M. Fréchet, La vie et l’œuvre d’Émile Borel, L’Enseignement mathématique 11 (1965).
3. Borel, Émile, Dictionary of Scientific Biography (via MacTutor).
4. Émile Borel, Wikipedia; Émile Borel, Wikipédia (French).
5. Émile Borel, Mathematics Genealogy Project.
6. Académie des sciences, Fonds Borel (archive inventory).

Mathematics

1. É. Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. ÉNS 12 (1895).
2. É. Borel, Leçons sur la théorie des fonctions, Gauthier-Villars, 1898.
3. É. Borel, Mémoire sur les séries divergentes, Ann. Sci. ÉNS 16 (1899); Leçons sur les séries divergentes, 1901.
4. É. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo 27 (1909).
5. É. Borel, La théorie du jeu et les équations intégrales à noyau symétrique gauche, C. R. Acad. Sci. 173 (1921); English translation in Econometrica 21 (1953).
6. M. Fréchet, Émile Borel, initiator of the theory of psychological games and its application, Econometrica 21 (1953); J. von Neumann, Communication on the Borel notes, Econometrica 21 (1953).
7. G. H. Hardy, Divergent Series, Oxford, 1949.
8. T. Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development, Chelsea, 1975.
9. J. von Plato, Creating Modern Probability, Cambridge, 1994.
10. R. Dimand and M. A. Dimand, The early history of the theory of strategic games from Waldegrave to Borel, History of Political Economy 24 (1992).