Infimum and Supremum of sets

Question

In one of my textbooks - A Probability Path, Resnick - the author defines $\inf_{k \ge n}A_k = \cap_{k=n}^\infty A_k$ and $\sup_{k \ge n}A_k = \cup_{k=n}^\infty A_k$. I'm having a hard time understanding the intuition behind these definitions. I believe the $\inf_{k \ge n}a_k$ where $a_n$ is a sequence of real numbers can be interpreted as the greatest lower bound of this sequence after an integer $n$. How does this interpretation relate to $\{\omega \in \Omega: \omega \in \cap_{k=n}^\infty A_k\}$ which means that $\omega$ is found in all events $A_k$? Also for the supremum- how does the union of $A_k$ translate to the least upper bound?

Answer 1

I believe the $\inf_{k \ge n}a_k$ where $a_n$ is a sequence of real numbers can be interpreted as the greatest lower bound of this sequence after an integer $n$.

The number $b_n = \inf_{k \geq n} a_k$ has the property that

1. $b_n \leq a_k$ for all $k \geq n$; (i.e., $b_n$ is a lower bound for $a_n, a_{n+1}, a_{n+2}, \dots$)
2. If $c \leq a_k$ for all $k \geq n$, then $c \leq b_n$. (i.e., there is no greater lower bound.)

The set $B_n = \bigcap_{k=n}^\infty A_k$ has the property that

1. $B_n \subseteq A_k$ for all $k \geq n$;
2. If $C \subseteq A_k$ for all $k \geq n$, then $C \subseteq B_n$.

In other words, $B_n$ satisfies the same conditions in terms of the relation $\subseteq$ that regular infimum does in terms of $\leq$.

Answer 2

The subsets of a set form a set partially ordered by the subset relation, that is $\subseteq$ plays the role of $\leq$. So the infimum is the largest set that is a subset of all the sets, namely their intersection. Similarly, the supremum is the smallest set that is a superset of all of them, namely the union.

Answer 3

Say $B=\bigcap_k A_k$. Then $B\subset A_k$ for every $k$, so $B$ is a lower bound for the $A_k$ in the sense of inclusion. And if $S\subset A_k$ for all $k$ then $S\subset B$; so $B$ is the greatest lower bound.