Let $\mu$ be the Lebesgue measure on ($\mathbb{R}$, $\mathbb{B_R})$. Define a Borel measure as follows:
$\displaystyle v(E) = \int_E \frac{1}{|x|} d\mu(x)$.
How do I prove that $v$ is $\sigma$-finite? I am confused on how to compute $v$ in general. For instance, what would $v(\{0\})$ be? Is it undefined because $\frac{1}{|x|}$ is undefined at $x = 0$?
The most obvious decomposition would be the union of $[-i, i]$, but I am quite sure that this is not finite because as $x$ grows small, $\frac{1}{x}$ grows large.