# Definition:Random Variable/Real-Valued/Definition 1

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## Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

A **real-valued random variable** on $\struct {\Omega, \Sigma, \Pr}$ is a $\Sigma$-measurable function $X : \Omega \to \R$.

That is, a function $X : \Omega \to \R$ is a **real-valued random variable** if and only if:

- $X^{-1} \sqbrk {\hointl {-\infty} x} = \set {\omega \in \Omega : \map X \omega \le x} \in \Sigma$

for each $x \in \R$, where:

- $\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$
- $X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.

## Sources

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- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 5.1$: Distribution Functions

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 2001: Michael A. Bean:
*Probability: The Science of Uncertainty*: $\S 2.1$ - 2013: Donald L. Cohn:
*Measure Theory*(2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics