The transfinite mean

TL;DR

This paper introduces a generalized mean for transfinite sequences, demonstrating its existence in probability spaces and exploring limitations related to the club filter on .

Contribution

It defines a transfinite mean, proves its existence in probability spaces, and discusses limitations involving the club filter on .

Findings

Every probability space admits a transfinite sequence matching measure and frequency.

The club filter on  does not admit such a sequence of order type .

The work extends classical notions of mean to transfinite contexts.

Abstract

We define a generalization of the arithmetic mean to bounded transfinite sequences of real numbers. We show that every probability space admits a transfinite sequences of points such that the measure of each measurable subset is equal to the frequency with which the sequence is in this subset. We include an argument suggested by Woodin that the club filter on $\omega_1$ does not admit such a sequence of order type $\omega_1$.