Let $f$ be a nonnegative Lebesgue measurable function on $[0,\infty)$ such that $\int_0^{\infty}f(x)dx < \infty$. Show that there exists a positive, non-decreasing, Lebesgue measurable function $\Phi$ on $[0,\infty)$ with $\lim_{x\rightarrow\infty} \Phi(x) = \infty$, and such that $\int_0^{\infty} \Phi(x)f(x)dx < \infty$.
This is a past prelim problem that I'm working on. I'm really unsure how to proceed with it and I'm looking for some help. I feel like using the monotone convergence theorem might be a possibility because $\Phi$ is supposed to be non-decreasing but beyond that I'm unsure.
If $a_n\geq0$ for all $n$ and $\sum a_n<\infty$. There exists a strictly positive strictly increasing sequence $b_n$ such that $b_n \to \infty$ and $\sum a_nb_n <\infty$. Let $\Phi (x)=b_n$ on $[n,n+1)$ for $n=0,1,2...$ where $a_n=\int_n^{n+1}f(x)\, dx$. [One explicit choice $b_n$ is $(a_n+a_{n+1}+...)^{-1/2}$. This is an exercise in baby Rudin].