# Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E \frac{|X-Y|}{1+|X-Y|}$)).

I want to know are there any good characteristics about the relatively compact sets of this space.

It's pretty well known that the Ky Fan metric induces the topology of convergence in measure (or convergence in probability). (Of course it should be regarded as a metric / topology on the space $L^0(P)$ of equivalence classes of random variables, mod $P$-almost sure equality.)

In V. Bogachev's book Measure Theory, Exercise 4.7.129 is as follows:

4.7.129. (Fréchet [316], Veress [974]) Let $\mu$ be a probability measure on a space $X$ and let $M$ be some set of $\mu$-measurable functions. Prove the equivalence of the following conditions:

1. the set $M$ has compact closure in the metric of convergence in measure (Exercise 4.7.60);

2. every sequence in $M$ contains an a.e. convergent subsequence;

3. for every $\varepsilon > 0$ and $\alpha > 0$, there exists a finite collection of measurable functions $\psi_n, \dots, \psi_n$ such that, for every function $f \in M$, one can find an index $i \le n$ with $\mu(x \colon |f(x) - \psi_i(x)| \ge \varepsilon) < \alpha$;

4. for every $\varepsilon > 0$, there exist a number $C > 0$ and a finite partition of the space into disjoint measurable parts $E_1, \dots, E_n$ such that, for every function $f \in M$, there exists a measurable set $E_f$ with the following properties: $$\mu(E_f) < \varepsilon, \quad \sup_{x \in X \setminus E_f} |f(x)| < C, \quad \sup_{x,y \in E_i \setminus E_f} |f(x) - f(y)| < \varepsilon$$ for all $f \in M$ and $i = 1,\dots ,n$.

Hint: see Dunford, Schwartz [256, Theorem IV.11.1].

The relevant references are:

• [256] Dunford N., Schwartz J.T. Linear operators, I. General Theory. Interscience, New York, 1958; xiv+858 pp.

• [316] Fréchet M. Sur les ensembles compacts de fonctions mesurables. Fund. Math. 1927. V. 9. P. 25–32.

• [974] Veress P. Uber Funktionenmengen. Acta Sci. Math. Szeged. 1927. V. 3. P. 177–192.

• Incidentally, the 3-digit reference numbers are not a typo. Bogachev's books feature a staggering number of references. Volume 1 has 1055 references and Volume 2 has 2038 (!!!) Mar 20, 2015 at 3:45