question on the meaning of supremum I know that supremum means least upper bound. If I have a sequence of events, $\{A_n\}_{n=1}^\infty$
then $$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$
I'm having trouble understanding this statement: 
"The supremum of a collection of elements in a partially ordered set is its least upper bound, so $\sup_{j\geq n} A_j$ should be a set and it should hold that $A_j \subset \sup_{j\geq n} A_j$ for all $j \geq n $. Because the supremum should also be the smallest upper bound it is not hard to see that $$\sup_{j\geq n} A_j= \bigcup_{j=n}^\infty A_j$$"
Why does $j \geq n $ matter? Also, I don't understand what set is partially ordered, and also why the supremum of $A_j$ is itself a set. Shouldn't it just be an element? And I also don't see how the supremum is the same thing as the union.

Probability events are (the measurable) subsets of the probability space $\Omega$, and the partial order among them is simply the relation 'being subset of'. So, for any countable set of events $B_i$, their supremum is just $\bigcup_i B_i$ (which is still measurable).
By the way, we also have
$$\limsup_n A_n\ =\ \lim_n\sup_{j\ge n} A_j\ =\ \bigcap_n\bigcup_{j\ge n} A_j$$
