I am trying to understand the relation between population/probability space/sampling in Econometrics. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the three concepts.
Consider a target population of individuals; for each individual in this population we believe that $$income=beta*education+u$$
where $income, education, u$ are real numbers. $u$ collects all the variables affecting income in addition to education.
We are interested in learning about $\beta$.
Sub-question 1: Imagine to extract at random an individual $m$ from the population and observe her income, education, and additional features (denoted by the random variables $Y_m, X_m, U_m$). Intuitively, I understand why we can say that the income, education, additional features of the extracted individual are random variables: we don't know a priori which individual will be picked from the urn representing the whole population and, hence, we attach a probability to each potential outcome. More formally, to define a random variable we need a probability space $(\Omega, \mathcal{F}, Pr)$. Is $\Omega$ set equal to the population?
Sub-question 2: If the answer to question 1 is YES, then, if we knew the entire population (i.e, if we knew $(\Omega, \mathcal{F}, Pr)$) and suppose $E(X_m^2)\neq 0$ and $E(X_mU_m)=0$, we could easily compute the exact value of $\beta$ as $$ \beta=E(Y_mX_m)/E(X_m^2) $$ Correct?
Sub-question 3: The problem is that we don't know the entire population (i.e, we don't know $(\Omega, \mathcal{F}, Pr)$) and so we try to approximate in some good way $E(Y_mX_m)$ and $E(X_m^2)$ by appropriately taking a subset of the whole population (sampling). For example, a way to appropriately take a subset of the whole population is the following: for $m=1,...,M$:
We draw at random an individual from the urn containing the entire population, we label him/her with the index $m$, and we register his/her income and education level (denoted by the random variables $Y_m, X_m)$. The additional features affecting income remain unobserved (denoted by the random variables $U_m$).
We put back in the urn individual $m$.
The sampling scheme just described implies that $$ (i) \hspace{1cm}\{Y_m, X_m, U_m\}_{m=1}^M \text{ are i.i.d. across $m$} $$
We then define $$ \hat{\beta}=\frac{\frac{1}{M}\sum_{m=1}^MY_mX_m}{\frac{1}{M}\sum_{m=1}^M X_m^2} $$ By $(i)$, $\frac{1}{M}\sum_{m=1}^MY_mX_m\rightarrow_p E(Y_mX_m)$ and $\frac{1}{M}\sum_{m=1}^M X_m^2\rightarrow_p E(X^2_m)$.
Hence, $$ \hat{\beta}\rightarrow_p \beta $$ Correct?
Sub-question 4: Suppose also that $U_m$ is a continuous random variable. Does this implicitly imply stating that the population is very large or a continuum or infinite?