The above is extracted from Rudin's Real and Complex Analysis. Here $m$ denotes the Lebesgue measure on $\Bbb R^k$. I have questions about the red box.
How do I have to apply Theorem 7.8 or Theorem 7.10 to get the result? Functions $f$ in Theorem 7.8 and Theorem 7.10 are both belong to $L^1$, while if $m(E)=\infty$, we don't have $f=\chi_E \in L^1(\Bbb R^k)$.
Let $E_N=E \cap \{y: \|y\| <N\}$. Then $lim_{r \to 0} \frac {m(E\cap B(x,r))} {m(B(x,r)}=lim_{r \to 0} \frac {m(E_N\cap B(x,r))} {m(B(x,r)}$ for any $x$ and $N$ sufficiently large, say $N>1+\|x\|$. Since Theorem 7.8 is applicable to $E_N$ in place of $E$ the result follows. (Some minor details are left but I hope you can finish the proof).