random variables is mutually independent if and only if Notice that it is not necessary here to require that the probability distribution factorizes for all possible subsets as in the case for events. This is not required because e.g. implies . The measure-theoretically inclined may prefer to substitute events for events in the above definition, where is any Borel set. That definition is exactly equivalent to the one above when the values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values in any measurable space (which includes topological spaces endowed by appropriate σ-algebras). For real valued random vectors Two random vectors and are called independent if where and denote the cumulative distribution functions of and and denotes their joint cumulative distribution function. Independence of and is often denoted by . Written component-wise, and are called independent if For stochastic processes For one stochastic process The definition of independence may be extended from random vectors to a stochastic process. Therefore, it is required for an independent stochastic process that the random variables obtained by sampling the process at any times are independent random variables for any . Formally, a stochastic process is called independent, if and only if for all and for all where Independence of a stochastic process is a property within a stochastic process, not between two stochastic processes. For two stochastic processes Independence of two stochastic processes is a property between two stochastic processes and that are defined on the same probability space . Formally, two stochastic processes and are said to be independent if for all and for all , the random vectors and are independent, i.e. if Independent σ-algebras The definitions above ( and ) are both generalized by the following definition of independence for σ-algebras. Let be a probability space and let and be two sub-σ-algebras of . and are said to be independent if, whenever and , Likewise, a finite family of σ-algebras , where is an index set, is said to be independent if and only if and an infinite family of σ-algebras is said to be independent if all its finite subfamilies are independent. The new definition relates to the previous ones very directly: Two events are independent (in the old sense) if and only if the σ-algebras that they generate are independent (in the new sense).