The Monte Carlo Integral of a Continuum of Independent Random Variables

Consider a continuum of independent and identically distributed random variables corresponding to the points of the unit interval [0; 1]. Known technical diffculties are complemented by showing directly that the random sample path is almost surely not a Lebesgue measurable function. This refutes the common claim that, because of some version of the law of large numbers, the integral of each sample path equals the common mean of each random variable. To obtain a valid and useful result, we apply to the continuum of random variables the Monte Carlo method of numerical integration based on limits as the sample size tends to infinity of empirical finite sample averages of the realized random values. The resulting Monte Carlo integral is almost surely a degenerate random variable concentrated on the mean. A suitably modified version works when the different indexed random variables are merely independent with cumulative distribution functions that are measurable w.r.t. the index. Further generalizations to Monte Carlo integrals of conditionally independent random variables result, under conditions discussed in Hammond and Sun (2008, 2021), in non-degenerate random integrals that are measurable w.r.t. the conditioning -algebra.