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The following definition of a real-valued random variable (evidently) uses the notation $\{ X \in B \}$ for the pre-image of $B$ under $X$:

Let $(\Omega, F, \mathbb{P})$ be a probability space. A random variable is a real-valued function $X$ defined on $\Omega$ with the property that for every Borel subset $B$ of $\mathbb{R}$, the subset of $\Omega$ given by

$$ \{X \in B \} = \{\omega \in \Omega: X(\omega) \in B \} $$

is in the $\sigma$-algebra $F$.

(The definition is more or less quoted from Steve Shreve’s text on stochastic calculus for finance.)

Is $\{ X \in B \}$ a common way to write the pre-image in the context of random variables? Is there some background or motivation for this notation?

ExaviusP
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    Yes, using $P(X\in B)$ for $P({\omega\in \Omega,:, X(\omega)\in B})$ is quite common, more or less like verbally you would refer to the probability that the result of a die roll is $6$ rather than to the probability of the set of instances where the result of the die roll is $6$. – Sassatelli Giulio Feb 22 '23 at 18:20
  • @SassatelliGiulio Makes sense. Thank you – ExaviusP Feb 23 '23 at 17:46
  • Here is discussion that touches on similar topics. https://math.stackexchange.com/questions/694548/formal-definition-of-a-random-variable?rq=1 – ExaviusP Feb 24 '23 at 19:12
  • Also this. https://math.stackexchange.com/questions/516284/definition-of-a-real-valued-random-variable – ExaviusP Feb 24 '23 at 22:55

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