# Real-Valued Random Variable

A Real-Valued Random Variable is a Measurable Function which the outcome is a set of real numbers.

## References

### 2019

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Random_variable#Real-valued_random_variables Retrieved:2019-8-31.
• In this case the observation space is the set of real numbers. Recall, $(\Omega, \mathcal{F}, P)$ is the probability space. For real observation space, the function $X\colon \Omega \rightarrow \mathbb{R}$ is a real-valued random variable if :

$\{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}.$

This definition is a special case of the above because the set $\{(-\infty, r]: r \in \R\}$ generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that $\{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r])$ .