It's a well known fact in probability theory that if $\lbrace X_i\rbrace$ is a family of random variables, $\phi : [0,\infty) \to [0,\infty)$ satisfies $$\lim_{x \to \infty} \frac{\phi(x)}{x} = \infty \quad (*) $$ and $\sup_i E\phi(|X_i|) < \infty$, then the family $\lbrace X_i\rbrace$ is uniformly integrable. I want to know whether there is a single function $\phi$ satisfying (*) such that $\sup_i E\phi(|X_i|) < \infty$ iff $\lbrace X_i\rbrace$ is uniformly integrable.
I suspect the answer is no, and that even a family of one absolutely continuous random variable should suffice. So I conjecture the following: given any $\phi$ satisfying (*), there exists a nonnegative function $f$ such that $$\int_0^\infty f(x)dx=1$$ $$\int_0^\infty x f(x)dx < \infty$$ $$\int_0^\infty \phi(x)f(x)dx = \infty.$$
But I can't see offhand what to use for $f$. Ideas?
Nb. This is not homework. It was going to be, but I thought I should be able to solve it myself before assigning it to my students :)