A random variable is a measurable function from a probability space into a measurable space known as the state space (Doob 1996). Papoulis (1984, p. 88) gives the slightly different definition of a random variable as a real function whose domain is the probability space and such that:
1. The set is an event for any real number .
2. The probability of the events and equals zero.
The abbreviation "r.v." is sometimes used to denote a random variable.