What is a non-decreasing sequence of sets? What is a non-decreasing sequence of sets and how come it can have a limit?
It appear in a probability theory book
 A: It means that they satisfy $A_n \subset A_{n+1}$.
The limit is not a limit in the $\epsilon$-$\delta$ sense. It just means
$\cup_{n=1}^\infty A_n$.
For example, take $A_n = [0,n]$, then $\cup_{n=1}^\infty A_n = [0, \infty)$.
From a probability perspective, there is a real limit associated with
these nested sets. Suppose $p$ is the probability measure.
Since we have $p A_n \le p A_{n+1} \le 1$, we see that the sequence is non 
decreasing and bounded above and so has a limit. In fact, it is not hard to show that $\lim_{n\to \infty} pA_n = p (\cup_{n=1}^\infty A_n)$.
Here is a rather contrived example:
Consider the following probability defined on
measurable subsets of the plane. Let $pA = {1 \over \pi} m(A \cap B(0,1))$. That is, given a subset $A$ of the plane, we measure the area of the overlap
of $A$ and the unit disc $B(0,1)$ and divide the result by $\pi$ to make
it a probability measure.
Let $A_n$ be a regular $2^n$ polygon inscribed in the unit circle and let
$A_{n+1}$ be chosen by adding points in between the existing points. Then
we have $A_n \subset A_{n+1}$, so they are non decreasing (in this case
they can be considered increasing since $A_n \neq A_{n+1}$). Furthermore,
it is clear that the area of the polygons approaches that of the circle,
so we have $pA_n \to 1$.
A: When you are talking about sets, and you say the sequence is "increasing" or "non-decreasing", it just means that you have the containment $A_{1} \subset A_{2} \subset A_{3} \subset \dots \subset A_{n} \subset A_{n + 1} \subset \dots$ (usually, these are all proper subsets if you say "increasing" while they would not all necessarily have to be proper if you say "non-decreasing").  
For example, you could take $A_{n} = \{1, 2, \dots, n \}$.  Then $A_{1} = \{1 \}$, $A_{2} = \{1, 2 \}$, $\dots$, and clearly $A_{1} \subset A_{2} \subset A_{3} \subset \dots$.  This sequence would be "increasing".
If you take $A_{1} = \emptyset$, $A_{2} = \emptyset$, $A_{3} = \{1 \}$, $A_{4} = \{1 \}$, and $A_{n} = \{ 1 \}$ for all $n \geq 5$, then we have $A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \dots$.  Since some of the containments are not proper, this sequence would be "non-decreasing".
