I have a problem about the following theorem:
If $\mu$ is a positive Borel measure on $\mathbb{R}^k$ and $\mu \bot m$ ($m$ is the Lebesgue measure), then $$ (D\mu)(x) = \infty $$
almost everywhere with respect to $\mu$, where $(D\mu)(x)$ is given by $$ \lim_{r\to 0} {\mu(B(x,r)) \over m(B(x,r))}$$
For those who do not have the book, my question is that is $\mu$ required to be regular?
For those who have the book, the proof is on p.143. My question is that if $\mu$ is not assumed to be regular, how does the following work? $$ \mu(K) < 3^kN/j \text{ for all compact $K\subset W_{j,N}$ } \Rightarrow \mu(W_{j,N}) < 3^kN/j $$
Thanks!
I guess that regularity of $\mu$ is not necessary. Since $W_{j,N}\subset\mathbb{R}^k$ is open, there exist a countable collection of compact sets $K_i\subset W_{j,N}$ such that $W_{j,N} = \bigcup_{i=1}^\infty K_i$. Define $A_n = \bigcup_{i=1}^nK_i$. Then, each $A_n$ is compact, $A_1\subset A_2\subset\cdots$, and $W_{j,N} = \bigcup_{n=1}^\infty A_n$. By Theorem 1.19, $\mu(A_n)\to\mu(W_{j,N})$ and $\mu(W_{j,N})\le 3^kN/j$. I did not figure out how to prove $\mu(W_{j,N})<3^kN/j$ yet but it is sufficient for the proof.