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We discuss coupling of violation of Bell's inequality and non-Kolmogorovnessof statistical data in the EPR-Bohm experiment. We emphasize that nonlocaltyand "death of realism" are only sufficient, but not necessary conditions ofnon-Kolmogorovness. There can be found other sufficient conditions ofnon-Kolmogorovness and, hence, violation of Bell's inequality. We find oneimportant source of non-Kolmogorovness by analyzing axiomatics of quantummechanics. We pay attention to the postulate (due to von Neumann and Dirac) onsimultaneous measurement of quantum observables given by commuting operators.This postulate is criticized as nonphysical. We propose a new interpretation ofthe Born-von Neumann-Dirac rule for calculation of the joint probabilitydistribution for such observables. It gets a natural physical interpretation byconsidering conditional measurement scheme. We use this argument (i.e.,rejection of the postulate on simultaneous measurement to motivatenon-Kolmogorovness of the probabilistic structure of the EPR-Bohm experiment.

Violation of Bell's Inequality and Postulate on Simultaneous Measurement of Compatible Observables