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Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory.
Doob built on work by Paul Lévy and, during the 1940's and 1950's, he developed basic martingale theory and many of its applications. Doob's work has become one of the most powerful tools available to study stochastic processes.
In 1953 he published a book which gives a comprehensive treatment of stochastic processes, including much of his own development of martingale theory. This book Stochastic Processes has become a classic and was reissued in 1964. He is also the author of a well known book on measure theory.
References elsewhere in this archive:
J L Doob was the President of the American Mathematical Society in 1963 - 1964. You can see a history of the AMS and a list of AMS presidents.
He was the American Mathematical Society Colloquium Lecturer in 1959. You can see a history of the AMS Colloquium and a list of the lecturers.
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