Sufficient conditions for finite mean of a non-negative random variable

Question

Consider a continuous random variable that takes only non-negative values. Let the cumulative distribution function be $F(\cdot)$. Consider the following condition: $$\lim_{x\rightarrow\infty} x(1-F(x))=0.$$ Is it sufficient for the mean to be finite? If not, is there any other sufficient condition for the finite mean? Thank you!

Answer 1

Recall that $E[X] = \int_0^\infty (1-F(x))\,dx$. So this shows that something like $F(x) = 1-1/(x \log x)$ for large $x$ is a counterexample.

This does however show that $\lim_{x \to \infty} x^p (1-F(x)) = 0$ for some $p>1$ would suffice.