In the autumn of 1952 Bauer took the examinations to qualify as a high school teacher in Bavaria, then in February of the following year he was awarded a doctorate, with distinction, from Erlangen for his thesis Reguläre und singuläre Abbildungen eines distributiven Verbandes in einen vollständigen Vektorverband, welche der Funktionalgleichung f (x ∨ y) + f (x ∧ y) = f (x) + f (y) genügen which he had written with Otto Haupt as his advisor. The work was published in the papers Eine Rieszsche Bandzerlegung im Raum der Bewertungen eines Verbandes (1953) and a paper with the same title as his thesis which was published in Crelle's Journal in 1955. He remained at Erlangen as an assistant, and he habilitated there in 1956. His main interests at this time were in measure and integration, and in the work submitted for his habilitation he studied an abstract Riemann integral, introduced by L H Loomis, from the point of view of the theory of Radon measure.
Bauer spent 1956-57 as a Research Fellow at the Centre National de la Recherche Scientifique, Paris, working with Gustave Choquet and Marcel Brelot. It was at this time that he became interested in potential theory and convexity theory, two areas to which he was to make major contributions over the rest of his career. In 1961 Bauer was appointed to the University of Hamburg where he was appointed director of the Institute of Actuarial Mathematics and Mathematical Statistics and together with Emil Artin, Lothar Collatz, Helmut Hasse, Emanuel Sperner and Ernst Witt became one of the directors of the Mathematics seminar at the University of Hamburg. However, he did not take up the appointment at Hamburg straight away, for he had already planned spending the period from August 1961 to April 1962 as Visiting Associate Professor at the University of Washington in Seattle. In Hamburg, Bauer replaced Leopold Schmetterer, who had moved to Vienna to succeed Johann Radon who had died in 1956. Bauer made a research visit to Paris in the spring of 1964. He has served as Dean of the Faculty of Sciences at the University of Hamburg in session 1964-65.
On 1 September 1965, Bauer became a full professor at the University of Erlangen. We note that the University of Erlangen had been renamed Friedrich-Alexander-University Erlangen-Nurnberg in 1961 after merging with the Nurnberg College of Economics and Social Sciences. He held this chair for 31 years until his retirement, but he also held visiting positions in a number of universities, some of which we have mentioned above, but the remainder include the University of Munich, the University of Washington, the Sorbonne, the California Institute of Technology, New Mexico State University, and Aarhus University.
Chatterji  writes:-
What obviously impresses any one reading any of the papers of Bauer is the clarity and precision of the presentation; these qualities are perceptible right from his very first publications. It is therefore not surprising that he has written several highly successful textbooks and very useful expository articles.We now look briefly at some of these 'highly successful textbooks' he published. The first of these Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie was published in 1964 while he was still at Hamburg. L L Helms writes in a review:-
This book is in two parts. The first part is a standard development of measure theory, containing three chapters dealing with measure theory, integration theory, and product measure spaces in that order. ... The second part of the book is devoted to probability theory. Generally speaking, only probability theory as it pertains to product measure spaces is discussed. ... The book is efficiently organized, with emphasis entirely on positive results.Lectures Bauer gave in the summer semester of 1965 at Hamburg were published as Harmonische Räume und ihre Potentialtheorie (1966) and further lectures, inspired by Dieudonné's book Foundations of modern analysis appeared as two separate texts Differential- und Integralrechnung. I, II (1966). These latter two texts give an excellent account of the basic concepts of analysis. In 1968 he published a second book with the title Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie . The first half of this volume consists of the 1964 text, and it is followed by a section on measure in topological spaces and on the Fourier transform. The final part of the text proceeds at a much faster pace and covers topics such as the central limit theorem, conditional expectation, martingales, and some topics in stochastic processes. An English version Probability Theory and Elements of Measure Theory was published in 1972. Further editions of the German text appeared, the third in 1978, and the fourth edition of 1991 was published with the title Wahrscheinlichkeitstheorie . Because of the great popularity the book enjoyed, an extensive reworking and expansion of the sections on probability appeared in English translation as Probability theory in 1996, with the same treatment was given to the sections of measure theory, published in English translation as Measure and integration theory in 2001.
Another book by Bauer also became a classic. This was Mass- und Integrationstheorie (1990) which provided an introduction to measure theory and the theory of integration. A second edition was published in 1992. A collaboration with Bernd Anger led to the publication of Mehrdimensionale Integration (1976) which developed (from a review by L Janos):-
... the theory of multidimensional Lebesgue integration as a tool for handling integrals involved in problems of analysis and mathematical statistics (the gamma function, the Gauss distribution function, potential theory, the volume of the n-dimensional sphere, etc.).Bauer has served as Editor of Inventiones mathematicae (1966-79), Mathematische Annalen , Expositiones Mathematicae, and Aequationes Mathematicae. He served as a member of the Board of the Mathematical Research Institute of Oberwolfach for sixteen years from 1966. He was honoured with election to the Bavarian Academy of Sciences in 1975, and was also elected to the Finnish Academy of Sciences, the Austrian Academy of Sciences, the Royal Danish Academy of Sciences and the German Academy of Natural Scientists Leopoldina. He was the chairman of Mathematics section of this Academy in 1991. He was a winner of the Bavarian Order of Merit and the Bavarian Maximilian's Order of Science and the Arts. The Charles University, Prague, awarded him their medal in 1987 and in 1992 awarded him an honorary doctorate. The University of Dresden also awarded him an honorary doctorate in 1994. He was for many years a member of the German Mathematical Society and served as its President during 1976-77.
Netuka writes :-
In 1974 Bauer was an invited speaker at the International Congress of Mathematicians in Vancouver. His contribution ti the Proceedings starts as follows: "In 1964 Pierre Jacquinot opened a colloquium on potential theory in Orsay, France, by comparing potential theory with a road intersection in mathematics. This was ten years ago. Meanwhile traffic has increased, and crossroads had to be converted into interchanges of highways - also in potential theory." The first part of the contribution addresses three aspects of classical potential theory: superharmonic functions, Newtonian kernel and potentials, Brownian motion. The role of balayage is emphasised. The second part reflects two main sspects of potential theory of the early seventies: harmonic spaces and Markov processes. The last part is devoted to Fuglede's theory of finely harmonic functions, including an application to asympototic paths for subharmonic functions.In April 1979 Bauer was a plenary speaker at the British Mathematical Colloquium which was held at University College, London. He spoke on Korovkin approximation and convexity.
In 1980 Bauer received the Chauvenet Prize from the Mathematical Association of America. The American Mathematical Monthly reported :-
The Board of Governors of the Mathematical Association of America voted to award the 1980 Chauvenet Prize to Professor Heinz Bauer for his paper "Approximation and Abstract Boundaries," which appeared in this Monthly in 1978. A certificate and monetary award in the amount of five hundred dollars were presented to Professor Bauer at the Business Meeting of the Association on January 6, 1980. The Chauvenet Prize is awarded for a noteworthy paper of an expository or survey nature published in English that comes within the range of profitable reading for members of the Association. ... Professor Bauer is involved in research in integration theory, functional analysis (convexity and approximation theory), potential theory, and Markov processes. ... The paper for which Professor Bauer received the Chauvenet Prize discusses three famous theorems of P P Korovkin that concern uniform approximation of functions. These theorems are presented in a well-chosen setting and are illustrated and illuminated superbly with a collection of examples and applications. The paper is accessible to graduate students who have learned about the Lebesgue integral.Bauer retired from his chair in Erlangen in the spring of 1996. Sadly he suffered a stroke in the summer of that year from which he never fully recovered. In addition to his outstanding professional qualities, Bauer was known for his exceptionally broad general education, his knowledge of literature and history, his love of music and his high appreciation of cultural values.
Article by: J J O'Connor and E F Robertson