Measure-theoretic Definition
The most formal, axiomatic definition of random variables 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 (Ω, ℱ, P) be a probability space and (E, ℰ) a measurable space. Then an (E, ℰ)-valued random variable is a function X: Ω→E which is (ℱ, ℰ)-measurable. The latter means that, for every subset B ∈ ℰ, its preimage X −1(B) ∈ ℱ where X −1(B) = {ω: X(ω) ∈ B}. This definition enables us to measure any subset B in the target space by looking at its preimage, which by assumption is measurable.
When E is a topological space, then the most common choice for the σ-algebra ℰ is to take it equal to the Borel σ-algebra ℬ(E), which is the σ-algebra generated by the collection of all open sets in E. In such case the (E, ℰ)-valued random variable is called the E-valued random variable. Moreover, when space E is the real line ℝ, then such real-valued random variable is called simply the random variable.
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