Alexandra Bellow

Alexandra Bellow (née Bagdasar; previously Ionescu Tulcea; born 30 August 1935) is a Romanian-American mathematician, who has made contributions to the fields of ergodic theory, probability and analysis.

Bellow was born in Bucharest, Romania, on August 30, 1935, as Alexandra Bagdasar.

Her parents were both physicians.

Her mother, Florica Bagdasar (née Ciumetti), was a child psychiatrist.

Her father, Dumitru Bagdasar, was a neurosurgeon.

During her marriage to Cassius Ionescu-Tulcea (1956–1969), she and her husband co-wrote many papers and a research monograph on lifting theory.

She received her M.S. in mathematics from the University of Bucharest in 1957, where she met and married her first husband, mathematician Cassius Ionescu-Tulcea.

She accompanied her husband to the United States in 1957 and received her Ph.D. from Yale University in 1959 under the direction of Shizuo Kakutani with thesis Ergodic Theory of Random Series.

After receiving her degree, she worked as a research associate at Yale from 1959 until 1961, and as an assistant professor at the University of Pennsylvania from 1962 to 1964.

Lifting theory, which had started with the pioneering papers of John von Neumann and later Dorothy Maharam, came into its own in the 1960s and 1970s with the work of the Ionescu Tulceas and provided the definitive treatment for the representation theory of linear operators arising in probability, the process of disintegration of measures.

In the early 1960s she worked with C. Ionescu Tulcea on martingales taking values in a Banach space.

In a certain sense, this work launched the study of vector-valued martingales, with the first proof of the ‘strong’ almost everywhere convergence for martingales taking values in a Banach space with (what later became known as) the Radon–Nikodym property; this, by the way, opened the doors to a new area of analysis, the "geometry of Banach spaces".

These ideas were later extended by Bellow to the theory of ‘uniform amarts’, (in the context of Banach spaces, uniform amarts are the natural generalization of martingales, quasi-martingales and possess remarkable stability properties, such as optional sampling), now an important chapter in probability theory.

In 1960 Donald Samuel Ornstein constructed an example of a non-singular transformation on the Lebesgue space of the unit interval, which does not admit a \sigma–finite invariant measure equivalent to Lebesgue measure, thus solving a long-standing problem in ergodic theory.

A few years later, Rafael V. Chacón gave an example of a positive (linear) isometry of L_1 for which the individual ergodic theorem fails in L_1.

Her work unifies and extends these two remarkable results.

It shows, by methods of Baire category, that the seemingly isolated examples of non-singular transformations first discovered by Ornstein and later by Chacón, were in fact the typical case.

From 1964 until 1967 she was an associate professor at the University of Illinois at Urbana–Champaign.

In 1967 she moved to Northwestern University as a Professor of Mathematics.

Their Ergebnisse monograph from 1969 became a standard reference in this area.

By applying a lifting to a stochastic process, the Ionescu Tulceas obtained a ‘separable’ process; this gives a rapid proof of Joseph Leo Doob's theorem concerning the existence of a separable modification of a stochastic process (also a ‘canonical’ way of obtaining the separable modification).

Furthermore, by applying a lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly measurable function; this gives a one line proof of Phillips's classical theorem (also a ‘canonical’ way of obtaining the strongly measurable version).

We say that a set H of measurable functions satisfies the "separation property" if any two distinct functions in H belong to distinct equivalence classes.

The range of a lifting is always a set of measurable functions with the "separation property".

The following ‘metrization criterion’ gives some idea why the functions in the range of a lifting are so much better behaved.

Let H be a set of measurable functions with the following properties: (I) H is compact (for the topology of pointwise convergence); (II) H is convex; (III) H satisfies the "separation property".

Then H is metrizable.

The proof of the existence of a lifting commuting with the left translations of an arbitrary locally compact group, by the Ionescu Tulceas, is highly non-trivial; it makes use of approximation by Lie groups, and martingale-type arguments tailored to the group structure.

It was Ulrich Krengel who first gave, in 1971, an ingenious construction of an increasing sequence of positive integers along which the pointwise ergodic theorem fails in L_1 for every ergodic transformation.

The existence of such a "bad universal sequence" came as a surprise.

Alexandra's second husband was the writer Saul Bellow, who was awarded the Nobel Prize in Literature in 1976, during their marriage (1975–1985).

Alexandra features in Bellow's writings; she is portrayed lovingly in his memoir To Jerusalem and Back (1976), and, his novel The Dean's December (1982), more critically, satirically in his last novel, Ravelstein (2000), which was written many years after their divorce.

Beginning in the early 1980s Bellow began a series of papers that brought about a revival of that area of ergodic theory dealing with limit theorems and the delicate question of pointwise a.e. convergence.

This was accomplished by exploiting the interplay with probability and harmonic analysis, in the modern context (the Central limit theorem, transference principles, square functions and other singular integral techniques are now part of the daily arsenal of people working in this area of ergodic theory) and by attracting a number of talented mathematicians who were very active in this area.

One of the two problems that she raised at the Oberwolfach meeting on "Measure Theory" in 1981, was the question of the validity, for f in L_1, of the pointwise ergodic theorem along the ‘sequence of squares’, and along the ‘sequence of primes’ (A similar question was raised independently, a year later, by Hillel Furstenberg).

This problem was solved several years later by Jean Bourgain, for f in L_p, p>1 in the case of the "squares", and for in the case of the "primes" (the argument was pushed through to p>1 by Máté Wierdl; the case of L_1 however has remained open).

The decade of the nineties was for Alexandra a period of personal and professional fulfillment, brought about by her marriage in 1989 to the mathematician Alberto P. Calderón.

Some of her early work involved properties and consequences of lifting.

Bourgain was awarded the Fields Medal in 1994, in part for this work in ergodic theory.

She was at Northwestern until her retirement in 1996, when she became Professor Emeritus.