Am I right in thinking that the weak law of large numbers, when stating that
$$\bar{X_n} \to \mu$$
in probability convergence is stating that the sequence of random variables $\{X_n\}$ tends to the random variable (as opposed to the real number) $M = \mu$, that is $M(\omega) = \mu \ \ \forall \omega$, and $p_M(\mu) = 1, \ p_M(x \neq \mu) = 0$?
The reason I ask is because in all the statements of the law I've seen it simply states that it tends to the expected value $\mu$, and I think convergence in probability is only defined as convergence of sequences of RVs to RVs. It's a subtle and rather unimportant distinction but I want to make sure I have the concepts right. Thanks!