On a definition of the integral of a random function

The integral of a measurable random function xi(x) with respect to a nonnegative measure m is considered. It is shown that such an integral may be introduced as a limit in the probability of integrals of simple random functions. An analogue of the Lebesgue theorem for convergence in probability integral xi(n)(x) dm, n --> infinity is proved. A criterion that this random measure may be represented in the form of such an integral is obtained.