I'm trying to get a good handle on analysis counterexamples as they relate to convergence in $M(X)$ and $C_0(X)$. Awhile back there was an excellent discussion of pointwise convergence, convergence in $L^p$ norm, weak convergence in $L^p$ and convergence in measure, here.
How about a similar set of counterexamples for $M(X)$ and $C_0(X)$? Here we have the notions of vague convergence, weak* convergence, convergence in norm (where the norm of a complex Radon measure is its total variation).
How about these counterexamples (asked in Folland or a variation therein):
a) $\mu_n\to 0$ vaguely, but $\|\mu_n\|=|\mu|(X)|\nrightarrow 0$.
b) $\mu_n\to 0$ vaguely, but $\int f\ d\mu_n\nrightarrow \int f\ d\mu$ for some bounded measurable $f$ with compact support.
c) $\mu_n\ge 0$ and $\mu_n\to 0$ vaguely, but $\mu_n((-\infty,x])\nrightarrow \mu((-\infty,x])$ for some $x\in\mathbb{R}$.
d) $\{f_n\}\in C_0(X)$ converges weakly to some $f$, but not pointwise.
Any links to conceptual ways of internalizing these different notions of convergence, in addition to providing counterexamples, would be greatly appreciated.
Thanks.
EDIT: Let me define the notions of convergence as Folland does:
Vague convergence means convergence with respect to the vague topology on $M(X)$, which is also known as the weak* topology on $M(X)$, which means that $\mu_\alpha\to \mu$ iff $\int f\ d\mu_\alpha\to \int f\ d\mu$ for all $f\in C_0(X)$.
Weak convergence means that convergence on $X$ with respect to the topology generated by $X^*$.
The norm on $M(X)$ is given by the total variation, so that $\|\mu\|=|\mu|(X)$, so that convergence with respect to this norm means that $|\mu_n-\mu|(X)\to 0$.
Incidentally I find it awkward working with the total variation in such discussions of convergence because there is no clear geometry to work with as far as I can tell; the definition is rather too abstract for me at the moment.