I am reading a probability text that asks the following question.
If $X_1, X_2, \dots$ are independent $\mathrm{Ber}(p)$ random variables, where $0 < p < 1$, then define $T : \min\{k : X_k = 1\}$, and give a complete, measure theoretic proof of the fact that $T \sim \mathrm{Geo}(p)$.
I guess I don't really understand what this means. First of all, what even is the underlying probability space here? It seems that $T$ is taking integer values and $X$ is taking $0, 1$ values.
It can be a little confusing at first that in probability, we often don't mention the probability space $\Omega$ on which random variables are defined. This is because $\Omega$ often plays a secondary role in what we are interested in, and the idea is that any "suitable" $\Omega$ would do. So here, you are supposed to just assume that $X_1,X_2,...$ are an infinite sequence of random variables all defined on the same probability space $\Omega$. You then want to prove that the function $T: \Omega \to \mathbb{N}$ defined in terms of the $X_i$ is also a measurable function with the desired distribution.
(I should say that the probability space $\Omega$ also comes with a probability measure $\mathbb{P}$ and a sigma algebra $\mathcal{F}$.)