I have a question on the definition of a random variable. I have read the tickets in a box interpretation here and here, which helped me immensely, but I have some doubts for the case in which the population is infinite. In particular, I am confused on what is the relation among infinite population-continuous/discrete random variables-measurability-zero probability events.
Following the discussion in the linked answer, suppose that my population $\Omega$ is an infinite list of paper slips (a generic paper slip is denoted by $\omega$) and in each paper slip I write numbers, e.g.,
X Y epsilon
paperslip1 5 1 0.5
paperslip1 5 0 0.2
paperslip1 6 0 0.3
...
1): In order to understand the measurability condition in the definition of a random variable, we firstly need to say which measure we are using. We need a probability measure, which a denote by $P: \mathcal{\Omega}\rightarrow \mathcal{F}$, where $\mathcal{\mathcal{F}}$ is a $\sigma$-algebra on $\Omega$. How do we construct it? The first naively idea that comes to my mind is by using the counting measure, e.g., assuming $\{5\}\in \mathcal{F}$ $$ P(\omega \in \Omega \text{ s.t. } X(\omega)\in \{5\})\equiv\frac{\text{ # paper slips with $5$ written on them}}{\text{# paper slips}}=\frac{\text{some number, not necessarily finite}}{\infty}=0 \text{ or }\frac{\infty}{\infty} $$ This is clearly not a probability measure. Hence, using the counting measure is wrong when we have an infinite population. Correct?
2): If my arguments in 1) are correct, then we need to find another way to construct a probability measure. The second idea that comes to my mind is by using the length measure, e.g., assuming $\{5\}\in \mathcal{F}$ $$ P(\omega \in \Omega \text{ s.t. } X(\omega)\in \{5\})\equiv\frac{\text{ Length of paper slips with $5$ written on them}}{\text{Length paper slips}}=\frac{\text{some finite number}}{\text{some finite number}}\in [0,1] $$ We can easily verify that all the properties that a probability measure should have are satisfied. Correct? Also, is it true that the length measure is always finite?
3): If my arguments in 2) are correct, could you help me to understand what is $\mathcal{F}$? It seems to me that all events are length measurable.
4): Within this context, is it correct to define continuous random variables, as random variables for which the length of singleton sets in $\mathcal{F}$ is zero? And, conversely, discrete random variables, as random variables for which the length of singleton sets in $\mathcal{F}$ is strictly positive?
