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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?

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Notice that simple random variable takes finitely many values.

Consider geometric distribution where it can take arbitrary positive numbers, hence it can't be a simple random variable.

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