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I have been working through Terry Tao's Introduction to measure theory. A draft can be found here.

In section 6, The one sided Hardy-Littlewood maximal inequality is proved. It stated that if $f : \mathbb{R} \to \mathbb{C}$ is an absolutly integrable function and $\lambda > 0$ then $$ m ( \{ x \in \mathbb{R} : {\rm sup}_{h > 0} \frac{1}{h} \int_{[x,x+h]} \lvert f(t) \lvert dt \geq \lambda \} ) \leq \frac{1}{\lambda} \int _{\mathbb{R}} \lvert f(t) \lvert dt.$$

In exercise 1.6.13 of the same section, it is claimed that this inequality is actually an equality. But if we let $\lambda = 1$ and $f = \frac{1}{2} \chi_{[0,1]}$ then the LHS = $0$ and the RHS = $\frac{1}{2}$.

I don't think I am missing anything and I know that Terry is not missing anything so what is the deal?

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Just a guess: perhaps it is a typo in referencing, and it is really supposed to be the inequality in Exercise 1.6.12 that is an equality. (On the original webpage… where this appeared, this interpretation is what you would get if you replaced his reference to "Lemma 12" (which does not exist, but links to the one-sided HL maximal inequality, which is Lemma 9) with a reference to "Exercise 12". If this is indeed an error, someone should let Tao know; it is not yet in the errata for his book.) – leslie townes Apr 21 '12 at 5:26
yup, I agree. I will post a comment on his blog. – DBr Apr 21 '12 at 8:13
Just a remark: I won't like it, if it became common in this forum to call persons in the subject line (even if it is here made a bit "poetic"). This is often the beginning of trashing the forum and of getting it converted to some (mostly unpleasant) personal exchange. Might you consider to adapt that subject line? – Gottfried Helms Apr 21 '12 at 9:06
yeah, sorry I couldn't resist! I will edit the title accordingly – DBr Apr 21 '12 at 23:28
Very kind! Thank you! – Gottfried Helms Apr 22 '12 at 6:03

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