I need to find some set of (minimal) conditions to put on a family of probability density functions with bounded support so that the family becomes compact. (I want to use Sion's theorem, which requires the compactness condition). In other words, I need to define $\textsf{conditions}$ so that the family $$\mathcal{F}=\{f: f \textrm{ satisfies}\textsf{ conditions}\} $$ becomes compact. A natural set of conditions that I came up with is:
- $f$ is a PDF with bounded support $[0,K]$ (by definition).
- There exists a fixed $U$ such that $\forall f\in \mathcal{F}$, $f$ is discontinuous at most in $U$ points.
- There exists a fixed $U'$ so that $\forall f\in \mathcal{F}$ and $\forall x\in[0,K]$ we have $f(x)<=U'$.
I think $\mathcal{F}$ should be compact under these conditions, but I'm not sure about the best way to prove it, one way that I have in mind is proving that $\mathcal{F}$ is totally bounded in some metric space.
Also, I want to see if we can allow the members of $\mathcal{F}$ to have discontinuities defined by the dirac delta function? I don't have any proof ideas for this latter relaxation yet. Your ideas are very much appreciated!