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Doeblin also contributed to the theory of random chains with complete connection, some of which was used in a paper by him on ergodic properties of continued fractions.
His ergodic theorem transformed the Maxwell - Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory.
His ergodic theorem gave the kinetic theory of gases a rigorous basis.
He introduced ergodic methods into number theory in his first work.
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