I would like to confirm my basic understanding of how a probability space, random variables, and covariance should be 'meaningfully' defined in the context of experimentally acquired data through a simple example.
Suppose I have $3$ subjects whose names are: $A$, $B$, $C$. For each subject I collect height information and weight information.
Is the following interpretation correct? If not, could someone please provide the correct interpretation?
Let the scenario be 'encoded' by two random variables $H$ and $W$ defined relative to a common probability space $(\Omega,\mathcal F, P)$ where $\Omega=\{A,B,C\}$ and $\mathcal F=\{\{\},\{A\},\{B\},\{C\},\{A,B\},\{A,C\},\{B,C\},\{A,B,C\}\}$.
Presumably, from the description of this experiment, each subject has an equal probability of being 'selected'...right? If so, it seems like $P(\{A\})=P(\{B\})=P(\{C\})=\frac{1}{3}$. Further, because each subject is measured independently relative to the other subjects, I assume that $P(\{A,B\})=P(\{A,C\})=P(\{B,C\})=\frac{2}{3}$.
The mapping rules associated with each random variable are then just the data that are experimentally measured (e.g. $W(A)=107$, $W(B)=123$, $W(C)=114$ and $H(A)=62$, $H(B)=68$, $H(C)=65$.
It follows, then, that the $\text{cov}(H,W)=\displaystyle \frac{1}{3}\sum_{i=1}^3\left[h_i-\text{E}(H)\right]\cdot\left[w_i-\text{E}(W)\right]$, where $\text{E}(W)=\frac{1}{3}\cdot 107+\frac{1}{3}\cdot123+\frac{1}{3}\cdot 114$ and $\text{E}(H)=\frac{1}{3}\cdot 62+\frac{1}{3}\cdot 68+\frac{1}{3}\cdot 65$
I am unsure if the sample space is as straightforward as $\Omega=\{A,B,C\}$. I have seen other interpretations of what a sample space should look like when $n$ measurements are taken. For example, here is a quote taken from a statistics book:
In statistical inference and decision theory, the data set $(x_1,\cdots,x_n)$ is viewed as an outcome of the experiment whose sample is is $\Omega=\mathbb R^n$ (https://link.springer.com/book/10.1007/b97553)
Given that my description has three observations that could be interpreted as being of the form $x_1=\{h_1,w_1\},x_2=\{h_2,w_2\}, $ and $x_3=\{h_3,w_3\}$, it is unclear to me if $\Omega=\mathbb R^3$ makes more sense for my sample space than $\{A,B,C\}$.
If this is a proper interpretation, then I am not certain how covariance is calculated because I do not know how to generate jointly distributed random variables under this new interpretation (which requires, by definition, a common probability space).