The definition of convergence in distribution:
Let $S$ be a seperable metric space and $(X_n)_{n\in\mathbb{N}}$ a $S$-valued sequence of random variables. Then $(X_n)_{n\in\mathbb{N}}$ converge in distribution to a $S$-valued random variable $X$ if
$(*) \ \mathbf{E}[f(X_n)]=\mathbf{E}[f(X)]$
for all continuous, bounded functions $f:S\to\mathbb{R}$.
My question: In my case is $S$ a polish space. I have shown $(*)$ for all measurable, bounded functions $f:S\to\mathbb{R}$. Is this enough for convergence in distribution?