Integration in function spaces arose in probability theory when a gen-
eral theory of random processes was constructed. Here credit is cer-
tainly due to N. Wiener, who constructed a measure in function space,
integrals-with respect to which express the mean value of functionals of
Brownian motion trajectories. Brownian trajectories had previously been
considered as merely physical (rather than mathematical) phe- nomena. A.
N. Kolmogorov generalized Wiener's construction to allow one to
establish the existence of a measure corresponding to an arbitrary
random process. These investigations were the beginning of the
development of the theory of stochastic processes. A considerable part
of this theory involves the solution of problems in the theory of
measures on function spaces in the specific language of stochastic pro-
cesses. For example, finding the properties of sample functions is
connected with the problem of the existence of a measure on some space;
certain problems in statistics reduce to the calculation of the density
of one measure w. r. t. another one, and the study of transformations of
random processes leads to the study of transformations of function
spaces with measure. One must note that the language of probability
theory tends to obscure the results obtained in these areas for
mathematicians working in other fields. Another dir, ection leading to
the study of integrals in function space is the theory and application
of differential equations. A. N.