The law of large numbers states for $X_1,...,X_n$ random variables with finite expectation, \begin{equation} \mathbb{P}\bigg(\lim_{n\to\infty}\frac{S_n}{n}=\mu\bigg) = 1 \end{equation} where $S_n = X_1+...+X_n$.
I'm struggling to understand the difference between this definition and
\begin{equation} \lim_{n\to\infty} \mathbb{P}\bigg(\frac{S_n}{n}=\mu\bigg) = 1. \end{equation}
Are they equivilant? Does one imply the other? I know in general that the probability of a limit and the limit of the probability are not equivilant but struggling to understand the reason in this case.
Essentially you are asking why the probability measure is not continuous. Note that given a sequence of events $A_n$,
$$\lim_{n \to \infty}\mathbb{P}(A_n)=\mathbb{P}\left(\lim_{n \to \infty }A_n\right)$$
holds if $A_n$ is monotonic, i.e. either $A_1\subseteq A_2\subseteq A_3...$ or $A_1\supseteq A_2\supseteq A_3...$
Without monotonicity, continuity need not hold.
Consider $X_1,X_2,X_3,...$ iid $Bern(1/\pi)$. SLLN tells us
$$P\left(\frac{1}{n}\sum_{i=1}^n X_i \overset{n\to \infty}{\to}1/\pi\right)=1.$$
However, since $\sum_{i=1}^n X_i\sim Bin(n,1/\pi)$ is integer-valued,
$$P\left(\frac{1}{n}\sum_{i=1}^n X_i =1/\pi\right)=P\left(\sum_{i=1}^n X_i =n/\pi\right)=0\overset{n\to \infty}{\not\to}1.$$