Interpretation of Kolmogorovs Law of Large Numbers and Convergence My lecture slides begin,

Consider a sequence of random numbers $X_1,X_2,...,X_n$ (the lecturer here stated
that random numbers and random variables - which in my understanding
are maps from a sample space to the real line - can be used
interchangeably) all sampled from the same distribution which has mean (expectation) $\mu$.
Then the arithmetic average  of the sequence
$S_n=\frac{1}{n}(X_1+X_2+...+X_n)\approx E[x]$
and
$S_n=\frac{1}{n}(X_1+X_2+...+X_n)= E[x]$ as $n\rightarrow \infty$

It then goes on to state almost surely convergence

$X_n \rightarrow X \leftrightarrow P(\omega\in\Omega\mid X_n(w)\rightarrow X(w)$ as $n\rightarrow \infty)=1.$

My question is a conceptual one, what do we mean by $X_n\rightarrow X$ and $X_n(w)\rightarrow X(w)$?
For that matter what do we mean by $X_n$?
Are these aprroximations to the distribution $X$ after the $n_{th}$ trial if so when summing $S_n$ are we summing "maps from a sample space to the real line"? How would one add a map?
Am I correct in reading almost sure convergence as "$X_n$ becomes the distribution $X$ if and only if the probability of an outcome is 1 such that $X_n$ evaluated at that outcome becomes the value of $X$ evaluated at that outcome as we take successive approximations of $X$, $X_n$"
Please help!

I take it that the random variables $X_1,X_2,\ldots$ are assumed to be independent - this is an important assumption.
For a given element of the sample space, $\omega$, the sequence $X_1(\omega),X_2(\omega),\ldots$ is a sequence of real numbers. Saying that $X_n(\omega)\to X(\omega)$ is shorthand for saying that the limit
$$
\lim_{n\to\infty}X_n(\omega)
$$
exists and equals $X(w)$. That is, for all $\omega$ and for all $\epsilon$, there exists a number $N(\omega,\epsilon)$ such that for all $n>N(\omega,\epsilon)$ we have $|X_n(\omega)-X(\omega)|<\epsilon$.
Saying $X_n\to X$ is shorthand for saying that $X_n(\omega)\to X(\omega)$ for all $\omega$ in the sample space.
Each $X_n$ is a random variable, that is, a map from the sample space to $\mathbb R$. However, we are also assuming that these random variables are independent. That means we are assuming that there is a common sample space $\Omega$ such that $X_1,X_2,\ldots$ are all functions from $\Omega$ to $\mathbb R$, meaning that for every $n$ and for every sequence of real numbers $a_1,\ldots,a_n$ the events $\{X_1<a_1\},\ldots,\{X_n<a_n\}$ are independent.
