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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?

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    More precisely about @robjohn 's comment: how does the book define measurable functions? Usual definitions I am familiar with requires that $\phi:\mathbb{R}^n \to\mathbb{R}$ such that the inverse image of Borel sets and Lebesgue. So by definition $\phi$ cannot equal $\infty$ at any point. (That $\lim_{x\to x_0} \phi(x) = +\infty$ is not the same as $\phi(x_0) = \infty$!)2011-11-17

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When dealing with functions $f, g:{\mathbb R}^n \rightarrow [0,\infty]$, in order to make Lebesgue integration theory work, you specify that if $f(x) = 0$ and $ g(x) = \infty$, then $f(x)g(x) = 0$. This is the only way you can make commutativity, associativity, and distributivity hold (i.e. $(f(x) + g(x))h(x) = f(x)h(x) + g(x)h(x)$ and so on.) If you have Rudin's "Real and Complex Analysis" there's a short discussion at the bottom of p.18 about this issue. By the way, measurable functions in this setting are defined to be functions for which $f^{-1}(a,\infty]$ are measurable for all $a$. Equivalently, $f^{-1}(\infty)$ and each $f^{-1}(a,\infty)$ is measurable.

So in the case you're worried about, your inequality reduces to $0 \leq 0$ which is obviously ok.

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    Excellent answer!2011-11-21