About the notation of the probability measures
Our textbook uses the following notation for weak convergence of probability measure:
$\mu_{n}\overset{w}{\rightarrow}\mu$
The relationship for convengence in distribution and weak convergence of ``distribution'' measures.
$X_{n}\overset{\mathcal{D}}{\rightarrow}X\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$.
I wonder whether $P^{X_{n}}$ and $\mu_{n}$ are the same thing, i.e. $\mu_{n}\overset{w}{\rightarrow}\mu\Leftrightarrow P^{X_{n}}\overset{w}{\rightarrow}P^{X}$
Thank you!