Example of sequence of measures? Can you give me some examples of sequence of measure that converge to a measure? 
(I am reading the topic of weakly convergence of measure, and convergence of random variables in distribution) 
 A: Convergence makes sense in metric spaces, so any metric on the space of measures will lead to a notion of convergence of measures.
For instance, consider the space of probability measures on $\mathbb{R}$. Define the total variation metric
$$
\|\mu-\nu\|_{TV}=\sup_{E\subset \mathbb{R}}|\mu(E)-\nu(E)|
$$
where $E$ ranges over measurable sets. It's not hard to see that this is a metric.
For example, consider the family of measures
$$
\mu_n(E)=\frac{1+\frac{1}{n}}{2}\int_E \exp\left[-|x|\left(1+\frac{1}{n}\right)\right]\ dx,\qquad E\subset \mathbb{R},n\in \mathbb{N}.
$$
After a bit of bounding integrals, we see that $\mu_n$ converges in this metric to
$$
\mu(E)=\frac{1}{2}\int_E e^{-|x|}\ dx,\qquad E\subset \mathbb{R}.
$$
A: Your question is a bit vaguely defined, but I'll try to give some hints. There are many, many types of convergence of measures, each of which is useful in different problems. In most of the cases, convergence is given by some metric, that is you say that 
$$
  \mu_n \stackrel{d}{\longrightarrow} \mu
$$
iff $d(\mu_n,\mu) \to 0$, where $d$ is one of the metrics on measures. There is at least a dozen different metrics used. Note, that here we talk about metrics on spaces of measures, not on the underlying spaces. That is, if we are talking about measures on $\Bbb R$, then $\Bbb R$ is the underlying space - it can be a metric space on itself, endowed e.g. with the Euclidean metric. Given a $\sigma$-algebra on $\Bbb R$, e.g. a Borel one $\mathscr B(\Bbb R)$ defined by the Euclidean metric on $\Bbb R$, we can talk about the space of, say, Borel probability measure on $\Bbb R$ - let's denote it by $\mathscr P(\Bbb R)$. This is a different space, and even if we fix the Euclidean metric on $\Bbb R$, we can introduce a lot of different metrics on $\mathscr P(\Bbb R)$.
One of these metrics is the total variation one, mentioned by pre-kidney. It's a very popular metric since if you know that $d_{TV}(\mu,\nu) = \epsilon$, then for any set $A$ you can say
$$
  \mu(A)-\epsilon \leq \nu(A) \leq\mu(A) + \epsilon.
$$
For example, imagine that $\nu$ is some tough measure, and you have to compute crazy integrals even to evaluate $\nu([0,1])$. In case you can approximate $\nu$ with some nice measure $\mu$ in total variation, you can get very useful bounds for $\nu([0,1])$. Not also, that a convergence in total variation does not depend on the metric/topology on the underlying space, only on the $\sigma$-algebra. A different example is given by the Kantorovich metric:
$$
  d_K(\mu,\nu) := \sup\left\{\int_\Bbb Rf\,\mathrm d\mu - \int_\Bbb Rf\,\mathrm d\nu: f\in \mathrm{Lip}(\Bbb R,\rho)\right\},
$$
where $\rho$ is some metric on $\Bbb R$ (e.g. Euclidean) and 
$$
  f\in\mathrm{Lip}(\Bbb R,\rho) \quad \iff \quad |f(x) - f(x')|\leq\rho(x,x') \quad \forall x,x'\in \Bbb R
$$
are non-expansive (Lipschitz) functions with respect to metric $\rho$. By changing $\rho$, the metric on the underlying space, you change the corresponding Kantorovich metric on the space of measures. Note the difference with the total variation metric which depends only on the $\sigma$-algebra of the underlying space, but not on the metric $\rho$ or even  the topology $\rho$ induces.
Now finally back to your topic. Weak convergence of measures is usually introduced not through some metric, but directly as follows:
$$
  \mu_n\Rightarrow \mu \quad \iff\quad \int_{\Bbb R}f\,\mathrm d\mu_n \to \int_{\Bbb R}f\,\mathrm d\mu \quad \forall f\in C_b(\Bbb R).
$$
In fact, in many cases the weak convergence of measures is metrizable by $d_K$, that is $\mu_n\Rightarrow \mu$ iff $d_K(\mu_n,\mu) \to 0$. Now, by contrasting it with the total variation case we can give some examples.
The main idea that you have to bear in mind is that weak convergence, or as we know now, convergence in Kantorovich metric, allows discrete things to converge to continuous things. For example, when you need to simulate Brownian motion, you run discrete Monte-Carlo simulations, but the converge weakly to the Brownian motion. They do not converge e.g. in the total variation metric cause the latter is super conservative. To emphasize this, think of a convergent sequence of reals $x_n\to x$. Then for the Dirac distribution we have $\delta_{x_n}\Rightarrow \delta_x$, but $d_TV(\delta_{x_n},\delta_x) = 1$ unless $x_n = x$.
A: A simpler example might be to define $\mu_n$ as the measure induced by a $U(0,1+1/n)$ variable, i.e.
$$\mu_n(E) = \frac{1}{1+1/n} \int_{E \cap [0,1+1/n]} dx$$
Then $\mu_n$ converges to the measure induced by a $U(0,1)$ variable, i.e.
$$\mu(E) = \int_{E \cap [0,1]} dx$$
Here $\mu_n$ converges to $\mu$ both weakly and in total variation. 
One thing that might help you is to note that weak convergence of measures is just setwise convergence: $\mu_n \to \mu$ weakly if and only if for every $E \in \mathcal{F}$ we have $\mu_n(E) \to \mu(E)$. (Here $\mathcal{F}$ is our $\sigma$-algebra.)
