Let $f:[0.1] \to \Bbb{R}$ such that $$f(x)=\sum_{n=1}^\infty \frac{1}{n^2}\frac{1}{\sqrt{|x-r_n|}}$$ where $\{r_1,r_2…r_n…\}=\Bbb{Q} \cap [0,1]$
We define the measure $$μ(Ε)=\int_E f^2 dx$$ for every Lebesgue measurable subset $E$ of $[0,1]$
I have to prove that this measure is sigma finite.
I already proved that: $$\mu <<m$$ $$f \in L^1([0,1])$$ $$\forall (a,b) \subseteq [0,1] \Rightarrow \mu((a,b)=+\infty$$
Can someone give me a hint to prove that this measure is sigma finite?
Thank you in advance.
Best Answer
$f \in L^{1}([0,1])$ implies that $|f| <\infty$ almost everywhere w.r.t $m$, hence w.r.t. $\mu$. The sets $\{x:|f(x)| \leq n \}$ have finite $\mu$-measure and their union differs from $[0,1]$ only by a $\mu$ null set. Hence $\mu$ is sigma finite.