Notation confusion in the Wikipedia article on the Law of Large Numbers

Notation confusion in the Wikipedia article on the Law of Large Numbers

In my infamous attempt at mastering (at my humble level) the "art" of
probability and statistical theory, I was reading the Wikipedia article on
the Law of Large Number and got confused by a couple of notations.
1 - mean vs expected value
First it is written:
$\bar X \rightarrow \mu$ for $n \rightarrow \infty$
For me $\mu$ is the "population" mean and not the expected value. The
definition of the law is that the sample mean converges to the random
variable expected value as the sample size approaches infinity (not its
population mean). I realize the population mean and the expected value are
equal but they are not computed the same way and I found this notation
misleading (because it doesn't follow directly the definition). What do
you think?
2 - random variable vs observation
Second it is written:
$\bar X = {1 \over n}(X_1 + X_2 + ... + X_n)$
"where X1, X2, ... is an infinite sequence of i.i.d. integrable random
variables with expected value $E(X1) = E(X2) = ...= \mu$."
Why I am confused is that for me the sample mean is computed as the
average of n observation where the observations are produced by a random
variable X. So it would be for me at least less misleading to use x lower
case (observation, realization) instead of X uppercase. Or is correct here
to use X and if so why?
EDIT: I understand you need to write $E[X_1]$ and can't write $E[x_1]$.
The expected value of an observation wouldn't make much sense. But that
what's the meaning really of $X_n$ in $\bar X = {1 \over n}(X_1 + X_2 +
... + X_n)$. For me a random variable is a function not really a number?
An average of functions?
But I am not an expert so there might be an explanation to both notations?
Thank you.