If $\ln x$ is integrable, then is $x \ln x$ also integrable?

Question

I have a very simple problem. Assume we have a finite measure $\mu$ on $[1,\infty)$, and \begin{align} \int_1^\infty t ~d\mu(t) < \infty. \end{align} My question is if this implies \begin{align} \int_1^\infty t \ln t ~d\mu(t) < \infty. \end{align} It would be nice if this is true but maybe someone sees a simple counterexample?

Answer 1

Example: $d\mu(t) = {dt\over t^2\log^{3/2}(t+1)}$.

Answer 2

Let $\mathrm{d}\mu = \frac{\mathrm{d}t}{t^\alpha \ln^\beta t}$. You should be able to show the two integrals converge if $\alpha > 2$ and, in the first case, $\beta < 1$, and in the second, $\beta < 2$. Surely you can find something in $(1,2]$.