The transfinite mean

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 ${\omega }_1$.