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Let ¡ii, J, P] be a probability space. The subset of Si on which an arbitrary sequence of random variables converges is shown to be equivalent to the intersection of three other sets, each specified by the almost sure convergence of a certain sequence of random variables. Kolmogorov's three-series theorem, which gives necessary and sufficient conditions for the almost sure convergence of a sequence of sums of independent random variables, is obtainable as a particular case of the present result.doi:10.1090/s0002-9939-1971-0277020-5 fatcat:q7ankdbfhbfbbf6lhalhvl7q44