- #1

Testguy

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## Homework Statement

Let S be the set of all cumulative distribution functions (cdf), as defined in basic statistics (defined below).

Along with a metric d, (S,d) will defined a metric space.

My goal is (with any metric) to make a compact subset from this set S. Does anyone have any idea on how I can possibly do that? Is there any "clever" metric that I can use to do this, or is maybe not possible? Any advice is very welcome.

*This is really not homework, but to me the question seemed to fit better in here.*

## Homework Equations

A cdf is defined as the set of all bounded, right-continuous, non-decreasing functions from the reals to [0,1] with [itex]\lim_{x \rightarrow \infty} F(x)=1[/itex] and [itex]\lim_{x \rightarrow -\infty} F(x)=0[/itex].

## The Attempt at a Solution

I have so far used the supremum (or is it called uniform?) metric [itex]d(F_1,F_2) = \sup_x |F_1(x) - F_2(x)|[/itex] to create a metric space, and restricted the set by defining [itex] S_r= \{F \in S: d(F,R)<r \}[/itex], where} [itex]R[/itex] is some pre-defined cdf.

[itex](S_r,d)[/itex] does however not seem to establish a compact space. I have shown that this space is complete, but it does not seem like this set is totally bounded - at least all my attempts on proving that it is has lead to counterexamples.

Thank you for any advice.