Inconsistency of Measure-Theoretic Probability

TL;DR

This paper identifies a fundamental inconsistency in measure-theoretic probability, revealing a contradiction that challenges its logical foundation, though it does not impact practical applications as it pertains only to idealized theoretical constructs.

Contribution

It uncovers a previously unnoticed contradiction in measure-theoretic probability, highlighting the need for a revised mathematical framework such as constructive mathematics.

Findings

Discovered a contradiction: 1/2 = 0 in measure-theoretic probability.

The contradiction is purely theoretical and does not affect practical applications.

Constructive mathematics avoids this inconsistency.

Abstract

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.