What does the sup function mean in the context of metrics for probability measures/distances/differences?

Question

I was studying different probability metrics and distances and came across the following source:

http://www.springer.com/cda/content/document/cda_downloaddocument/9781461448686-c1.pdf?SGWID=0-0-45-1368104-p174541769

At one point they define the Uniform/kolmogorov metric as:

$$\rho(X,Y) := sup\{ |F_X(x) - F_y(x) | : x \in R\}$$

I am guessing that they mean by F the probability distribution of F, but what I am a little confused is what the $sup$ function is suppose to be. What does sup mean?

I am specifically trying to understand what the uniform metric means, but I don't really know what the sup function is suppose to return.

Could you also provide a couple of examples of what $sup$ is suppose to return/give?

Answer 1

$\sup$ means supremum, which is the least upper bound of a set. So for your example,

$$\sup\{ |F_X(x) - F_y(x) | : x \in R\}$$

would return the least upper bound of $|F_X(x) - F_y(x) |$ for all $x\in{R}$.

Answer 2

Let $A = \bigcup_{n=1}^{\infty} (0, 5 - 1/n)$. Then $\sup (A) = 5$ by using the definitions, but $5 \notin A$.