# Question on an exercise from Terry Tao's blog

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.

• Welcome to Math Overflow. – haidangel Mar 2 at 9:42
• I'm a bit surprised a question of this level is positively received on MO. I think this is much better suited for MSE. – MathQED Mar 2 at 19:04
• @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. – Rixinner Mar 3 at 0:13
• I've solved this exercise, if any one need the solution, please tell me and I'll write it down here. – Rixinner Mar 3 at 9:23

## 1 Answer

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.