SLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges a.s. to $\mu$.
However, suppose instead we know that $X_1,...,X_n$ are iid and their sample average converges a.s. to a constant. Can we argue that $X_1$ has a finite mean and hence that such a constant must be the mean of $X_1$?
The motivation here is that the statistician may not have population information, including even existence of the first moment. However, if the statistician had the ability to empirically observe strongly limiting behavior of the sample average, perhaps this may be enough to conclude existence of the first moment.