Understanding the definition of a random variable

I’m working through a math stats book on my own (I’ve always wanted to learn it), but I’m getting confused about the definition of a random variable. The book says that a random variable is a function from the state space $\Omega$ into some space $T$. I understand this in terms of some simple examples: take a finite state space where each event has a probability. Then, given some $X$, we can easily compute $E(X)$ by mapping each event in $\Omega$ to $X(\omega)$ and so on.

But, here’s my problem: we also talk about “Normal random variables” or “Cauchy random variables” or … I having a hard time connecting those random variables to the functional definition. What is the state space $\Omega$? My first guess would be $\Omega=\mathbb{R}$, but that doesn’t seem right because $P(\Omega)=1$ and equal length intervals should have equal probability, right? That doesn’t work if $\Omega=\mathbb{R}$ though…

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First of all, a random variable is usually defined as a function $X: \Omega \to \mathbb{R}$. So for any possible event in the state space $\omega \in \Omega$, the random variable $X(\omega)$ assigns a real number to that event.

Strictly speaking, probabilities are defined for special sets of events in $\Omega$. They are not defined on the target space $\mathbb{R}$. So if we’re being precise, it doesn’t makes sense to ask “what is the probability of $X = 3$?” Instead, we should be asking “what is the probability of the set of events corresponding to $X=3$?”

But, there’s a catch. Random variables are not just any old functions. They are measurable functions from $\Omega \to \mathbb{R}$. This means that any set of values in the target space $\mathbb{R}$ corresponds to some set of events in $\Omega$, for which a probability has been defined. Therefore, because of this fact we can cut corners and refer to “the probability of $X = 3$,” even though it doesn’t exactly make sense.

Which brings us to your question. When we speak of “Normal” or “Cauchy” random variables, we are describing how the random variables assign probabilities to events in the state space $\Omega$. We are not actually describing the state space itself. When we say that, for a Normal r.v. $X$ that $P(X \leq 1) = 0.84$, we are really saying that “the probability of all events $\omega$ that $X$ maps to a real number $\leq$ 1 is equal to 0.84.” But these events themselves can be anything.

So in short, the answer is: it depends.