I looked at textbooks and Wikipedia and learnt that a random variable, by definition, is a measurable function $X$ from a space $\Omega$, equiped with a measure $\mathbb{P}$, that is first, we need a measure space $(\Omega,\mathcal{F},\mathbb{P})$. Then a random variable (whether it's continuous or discrete) is a measurable function $$X:\Omega\rightarrow \mathbb{R}.$$
Now, that's very clear. And I also studied and understood the definition of a measure space.
So, let's consider an experiment with only 2 outcomes: success or failure (let's imagine for example, winning the lottery). Assume that the probabilty of success is 0.0003. Now, consider a sequence of independent repetitions of this experiment. So if we let $N$ the total number of failures in this sequence before the first success, then $N$ is a random variable.
My question: if I would like to see (or formulate) $N$ in terms of a space $(\Omega,\mathcal{F},\mathbb{P})$ and $N$ then is a function. How can we do that? For example, what is a setting for $\Omega$?
Thanks so much for any help.
Let us consider the following theorem:
Theorem: If $(X_i,\mathcal{E_i},\mu_i)_{i \in \mathbb{N}}$ are all probability spaces than there exists the probability space product : $(X,\mathcal{E},\mu)= \times_i (X_i,\mathcal{E_i})$ which satisfies $\mu(A_1 \times ... \times A_k \times X_{k+1} \times ... )=\mu_1(A_1)...\mu(A_k)$ for all $A_k \in \mathcal{E_i}$ where $k \in \mathbb{N}$.
Thus in your case you can easly take $X_i=\{0,1\}$ and $\mu_i(\{1\})=p$, $\mu_i(\{0\})=1-p$, obtaining the space you were searching for.
This is not trivial, because for example it is not clear also which $\sigma$-algebra you should take from $X=\times_i X_i$ due to the fact that $X$ is not countable (It has the continuum cardinality).