The law of large numbers with a continuum of i.i.d. random variables (Q1060768)

There are two problems with the common argument that a continuum of independent and identically distributed random variables sum to a nonrandom quantity in ''large economies''. First, it may be unintelligible in that it may call for the measure of a nonmeasurable set. However, there is a probability measure, consistent with the finite-dimensional distributions, which assigns zero measure to the set of realizations having that difficulty. A second difficulty is that the ''law of large numbers'' may not hold even when there is no measurability problem.