Title: Inconsistency of Measure-Theoretic Probability and Random Behavior of Microscopic Systems,
Abstract: We report an inconsistency found in probability theory (also referred to as
measure-theoretic probability). For probability measures induced by real-valued
random variables, we deduce an "equality" such that one side of the "equality"
is a probability, but the other side is not. For probability measures induced
by extended random variables, we deduce an "equality" such that its two sides
are unequal probabilities. The deduced expressions are erroneous only when it
can be proved that measure-theoretic probability is a theory free from
contradiction. However, such a proof does not exist. The inconsistency appears
only in the theory rather than in the physical world, and will not affect
practical applications as long as ideal events in the theory (which will not
occur physically) are not mistaken for observable events in the real world.
Nevertheless, unlike known paradoxes in mathematics, the inconsistency cannot
be explained away and hence must be resolved. The assumption of infinite
additivity in the theory is relevant to the inconsistency, and may cause
confusion of ideal events and real events. As illustrated by an example in this
article, since abstract properties of mathematical entities in theoretical
thinking are not necessarily properties of physical quantities observed in the
real world, mistaking the former for the latter may lead to misinterpreting
random phenomena observed in experiments with microscopic systems. Actually the
inconsistency is due to the notion of "numbers" adopted in conventional
mathematics. A possible way to resolve the inconsistency is to treat "numbers"
from the viewpoint of constructive mathematics.