I’ve been trying to work the following problem:
Let $\mathbb{R}^+:=[0,\infty)$ and let $f,g:\mathbb{R}^+ \to \mathbb{R}^+ $ be two measurable functions such that $$\int_0^x g(t)dt\leq \int_0^x f(t)dt,$$ for all $x \in (0,+\infty)$. If $\phi:\mathbb{R}^+\to \mathbb{R}^+$ is a non-increasing measurable function, show that $$\int_0^{\infty} g(t)\phi(t)dt\leq \int_0^{\infty} f(t)\phi(t)dt.$$
This is a question from a qualifying exam in Purude University. I’ve been trying several things but nothing seems to work. I think this problem can be solved using Fubini\Tonelli theorem, but I don’t really know how to proceed.
Any help would be hugely appreciated!
If you modify $\phi$ on a countable set it becomes left-continuous. Now there is a positive measure $\mu$ so that $\phi(t)=\phi(0)+\mu([0,t))=\phi(0)+\int\mathbb 1_{[0,t)}\,d\mu$.