# Example of non-"simple" random variable

In these lecture notes, at page 4 Definition 1.12, I've stumbled upon this definition of "simple" random variable, which I cannot understand in it's utility. For practicity, I will also write here the definition of random variable.

Random variable. Let $$(\Omega, \mathcal{F})$$ be a measurable space. A function $$X : \Omega \rightarrow \mathbb{R}$$ is called measurable if for any $$B \in \mathcal{B}(\mathbb{R}), X^{-1}(B) \in \mathcal{F}$$. A real-valued random variable in a probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$ is a mesurable function from $$X : \Omega \rightarrow \mathbb{R}$$.

A random variable $$X$$ is called simple if there exist $$n>0, x_1, x_2, ... x_n \in \mathbb{R}$$ and $$A_1, A_2, \dots, A_n \in \mathcal{F}$$ which

$$X(\omega) = \sum_{k=1}^n x_k \mathbb{1}_{A_k}(\omega)$$

where $$\mathbb{1}_{A_k}$$ is the indicator function of the event $$\omega$$ being in the sets $$A_k$$.

To me, all the random variables I can think of, are "simple". What is an example of non-simple random variable?