axiomatic definition of a random variable involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. The measure-theoretic definition is as follows. Let be a probability space and a measurable space. Then an -valued random variable is a measurable function , which means that, for every subset , its preimage is -measurable; , where . This definition enables us to measure any subset in the target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of is a possible outcome, a member of is a measurable subset of possible outcomes, the function gives the probability of each such measurable subset, represents the set of values that the random variable can take (such as the set of real numbers), and a member of is a "well-behaved" (measurable) subset of (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability. When is a topological space, then the most common choice for the σ-algebra is the Borel σ-algebra , which is the σ-algebra generated by the collection of all open sets in . In such case the -valued random variable is called an -valued random variable. Moreover, when the space is the real line , then such a real-valued random variable is called simply a random variable. Real-valued random variables In this case the observation space is the set of real numbers. Recall, is the probability space. For a real observation space, the function is a real-valued random variable if This definition is a special case of the above because the set generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on