a modification of if for all the following holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be stochastically equivalent or equivalent. Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse. If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. The theorem can also be generalized to random fields so the index set is -dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. Indistinguishable Two stochastic processes and defined on the same probability space with the same index set and set space are said be indistinguishable if the following holds. If two and are modifications of each other and are almost surely continuous, then and are indistinguishable. Separability Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space, which means that the index set has a dense countable subset. More precisely, a real-valued continuous-time stochastic process with a probability space is separable if its index set has a dense countable subset and there is a set of probability zero, so , such that for every open set and every closed set , the two events and differ from each other at most on a subset of . The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be -dimensional Euclidean space. The concept of separability of a stochastic process was introduced by Joseph Doob,. The underlying idea of separability is to make a countable