Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following:
*For every $1 , there is a constant $c_p>0$ such that, for any measurable functions on $\mathbb R^n$, $\phi\geq0$ and $f$, we have the inequality: (2.13) $\int\limits_{\mathbb R^n}\left(Mf(x)\right)^p\phi(x)dx\leq C_p\int\limits_{\mathbb R^n}|f(x)|^p(M\phi)(x)dx$* In the proof of this theorem the authors said that: when $M\phi(x)=\infty$ a.e., then (2.13) holds trivially. If $|f(x)|>0$ on a subset with positive measure, everything seems right. But if I chose namely $f(x)=0$ a.e., I do not see why (2.13) is trivial, because in this sense it may be $0\cdot \infty$ is undefined. So how should I understand (2.13)? Is it really trivial in that case and why?