Reference for a good multidimensional portmanteau theorem

Question

I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions:

1. The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$;
2. $E \{f(X_n)\} \to E\{f(X)\}$ for every bounded continuous function $f$;
3. $E\{f(X_n)\} \to E\{f(X)\}$ for every bounded Lipschitz function $f$;
4. $E \{\exp (i\; t\cdot X_n)\} \to E\{\exp(i\;t\cdot X)\}$ for every $t\in \mathbb R^\nu$ (Lévy's continuity theorem);
5. $\limsup_{n\to\infty} E\{f(X_n)\} \leq E\{f(X)\}$ for every upper semi-continuous function $f:\ \mathbb R^\nu\to \mathbb R$ bounded from above;
6. $\liminf_{n\to \infty} E\{f(X_n)\} \geq E\{f(X)\}$ for every lower semi-continuous function $f:\ \mathbb R^\nu\to \mathbb R$ bounded from below;
7. $\varphi_n (x) \to \varphi(x)$ at every point $x$ where $\varphi$ is continuous, $\varphi$ being the cumulative distribution of $X$, and $\varphi_n$ that of $X_n$;
8. Assuming $X_n = (I_\Omega, \mu_n)$, $\mu_n(A)\to \mu(A)$ for every continuity set $A$ of $\mu$, that is, for every Borel set $A$ for which $\mu(\partial A) = 0$.

Assertions 1-3, 5-6 and 8 are in this Wikipedia article, and are probably part of any exposition. I believe it should not be too difficult to find a n-d version of Levy's theorem (would appreciate though). But the equivalence with assertion 7 seems to entail using a multidimensional version of the cumulative distribution of a random vector and a n-d Stieltjes integral. This does not seem to be very classical. The best I have found is this article which does not even define the n-d cumulative function and only seems to evoke assertion 7.