  We reveal a contradiction in measure-theoretic probability. The contradiction
is an "equation" $1/2 = 0$ with its two sides representing probabilities.
Unlike known paradoxes in mathematics, the revealed contradiction cannot be
explained away and actually indicates that measure-theoretic probability is
inconsistent. Appearing only in the theory, the contradiction does not exist in
the physical world. So practical applications of measure-theoretic probability
will not be affected by the inconsistency as long as "ideal events" in the
theory (which will never occur physically) are not mistaken for real events in
the physical world. Nevertheless, the inconsistency must be resolved.
Constructive mathematics can avoid such inconsistency. There is no
contradiction reported in constructive mathematics.
