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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 terrytao.wordpress.com/2010/10/16/… 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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