Abstract
The convergence of the generalized derivatives of a sequence of finitely additive set functions along a sequence of algebras is proved very simply by using only the maximal inequality. This gives a purely measure-theoretical proof and extension of the semi-martingale convergence theorems.
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References
CARATHÉODORY, C.: Algebraic theory of measure and integration, New York: Chelsea 1963.
CHATTERJI, S. D.: Martingale convergence and the Radon-Nikodym theorem in Banach spaces. Math. Scand.22, 21–41 (1968).
DOOB, J. L.: Stochastic processes, New York: Wiley 1968.
DUNFORD, N., and SCHWARTZ, J. T.: Linear operators I, New York: Interscience 1958.
HEWITT, E., and YOSIDA, K.: Finitely additive measures. Trans. Amer. Math. Soc.72, 46–66 (1952).
JERISON, M.: Martingale formulation of ergodic theorems. Proc. Amer. Math. Soc.10, 531–539 (1959).
JOHANSEN, S., and KARUSH, J.: On the semi-martingale convergence theorem. Ann. Math. Statist.34, 690–694 (1966).
KRICKEBERG, K., and PAUC, C.: Martingales et dérivation. Bull. Soc. Math. France91, 455–544 (1963).
MUNROE, M. E.: Introduction to measure and integration, Reading: Addison-Wesley 1953.
RIESZ, F., and Sz.-NAGY, B.: Functional analysis, New York: Frederick Ungar 1955.
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Chatterji, S.D. Differentiation along algebras. Manuscripta Math 4, 213–224 (1971). https://doi.org/10.1007/BF01190277
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DOI: https://doi.org/10.1007/BF01190277