A paradox of measure-theoretic probability

Abstract

We report a paradox of measure-theoretic probability. Denote by B(R) and B(R) the - algebras corresponding to the real line R and extended real line R, respectively. Let R be equipped with a topology induced by a fixed metric. Let ; ; n; n; n = 1; 2; be probability measures on (R; B(R)), such that ; 1; 2; are distributions of i.i.d. random variables X;X1;X2; , respectively, with (f0g) = (f1g) = n(f0g) = n(f1g) = 1/2 and (f 1g) = n(f 1g) = 0, and n are distributions of Zn = maxfi : Xi = Yn; i 2 f1; 2; ; ngg where Yn = maxfXi : i 2 f1; 2; ; ngg. The weak limit of n is trivially , i.e., n ) ; n ! 1. Although n(f 1g) = 0 for all n, since (Zn)n 1 is non-decreasing, n ) where (f1g) = 1. So ( n)n 1 is not tight on (R; B(R)). As a probability measure on (R; B(R)), is the Dirac measure concentrated at f1g. However, when restricted to B(R), is a subprobability measure on (R; B(R)) with (R) = 0. Tightness of probability measures on (R; B(R)) is a condition to prevent mass from ‘escaping to infinity’. Although ( n)n 1 is trivially tight on (R; B(R)), the definition of Zn implies that ( )n 1 is intrinsically connected to ( n)n 1. We show that, from such connection, (f0g) = (R) = 0 is deducible, though it contradicts (f0g) = 1/2. See http://www.eee.hku.hk/research/doc/tr/TR-2014-001.pdf for more details.