Explanation of notation: a probability space equipped with measure P( . ) In a lecture I attended today, the professor made an off-hand comment of: "Suppose we have the set $S_n$ of permutations of $\\{1, 2, ..., n\\}$, which we can think of as a probability space equipped with measure $P( . )$." I'm not sure what this means - does it mean we have a probability of picking a random permutation with some probability, or something different...?

Any finite set $S$ can be equipped with a natural probability measure $P\ $ by setting, for any subset $A\subseteq S$,
$$P(A)={\mbox{number of elements in }A\over \mbox{number of elements in }S}.$$

This corresponds to selecting an item from $S$ _uniformly_ or _at random_. I suspect that your professor was thinking of applying this idea to the set of permutations $S_n$.