I've been reading Tao's An introduction to measure theory,a draft can be found here.An exercise from it is

Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely integrable function, and let ${f^*: {\bf R} \rightarrow {\bf R}}$ be the one-sided signed Hardy-Littlewood maximal function

$$\displaystyle f^*(x) := \sup_{h>0} \frac{1}{h} \int_{[x,x+h]} f(t)\ dt.$$

Establish the rising sun inequality

$$\displaystyle \lambda m( \{ f^*(x) > \lambda \} ) \leq \int_{x: f^*(x) > \lambda} f(x)\ dx$$

for all real ${\lambda}$ (note here that we permit ${\lambda}$ to be zero or negative).

I've prove it. But there it's another exercise after it

Exercise 31 Show that the left and right-hand sides in Exercise 30 are in fact equal when ${\lambda>0}$.

I tried to solve it and failed. And I found that if we take $f=1_{[0,1]}$, and $\lambda=\frac{1}{2}$ then the left-hand side is $\frac{1}{2}$ but the right-hand side is $1$, which are not equal.

Am I make any mistake? Please tell me.

Thanks in advance.

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    $\begingroup$ Welcome to Math Overflow. $\endgroup$ – haidangel Mar 2 at 9:42
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    $\begingroup$ I'm a bit surprised a question of this level is positively received on MO. I think this is much better suited for MSE. $\endgroup$ – MathQED Mar 2 at 19:04
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    $\begingroup$ @MathQED:When I googled this problem, I found a similar question on MSE without any answer. And the comment under it said questions like this should be posted on MO.So I posted it here. $\endgroup$ – Rixinner Mar 3 at 0:13
  • $\begingroup$ I've solved this exercise, if any one need the solution, please tell me and I'll write it down here. $\endgroup$ – Rixinner Mar 3 at 9:23

If $f=1_{[0,1]}$, then for real $x$ we have $$f^*(x)=\frac1{1-x}1(x<0)+1(0\le x<1).$$ So, for $\lambda=1/2$ the left-hand side is $$\tfrac12\,m(\{x\colon f^*(x)>\tfrac12\})=\tfrac12\,m((-1,1))=1,$$ which is the same as the right-hand side.


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