in the same mathematical space , which must be measurable with respect to some -algebra . In other words, for a given probability space and a measurable space , a stochastic process is a collection of -valued random variables, which can be written as: Historically, in many problems from the natural sciences a point had the meaning of time, so is a random variable representing a value observed at time . A stochastic process can also be written as to reflect that it is actually a function of two variables, and . There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a -valued random variable, where is the space of all the possible functions from the set into the space . However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined. Index set The set is called the index set or parameter set of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set the interpretation of time. In addition to these sets, the index set can be another set with a total order or a more general set, such as the Cartesian plane or -dimensional Euclidean space, where an element can represent a point in space. That said, many results and theorems are only possible for stochastic processes with a totally ordered index set. State space The mathematical space of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, -dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take. Sample function A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if is a stochastic process, then for any point , the mapping is called a sample function, a realization, or, particularly when is interpreted as time, a sample path of the stochastic process . This means that for a fixed , there exists a sample function that maps the index set to the state space . Other names