I want to show that the lebesgue measure is $\sigma$-finite on the following;
$C_\mathrm{open} = \{ A \subset \mathbb{R} : A\,\,\mathrm{open}\}$
$C_\mathrm{closed} = \{ B \subset\mathbb{R}: B\,\,\mathrm{closed}\}$.
Usually, for example for $C = \{(a,b): -\infty\leq a\leq b\leq \infty \}$ I took the interval $(-i,i)$ and used infinite unions etc to write it as half open so that I could take the lebesgue measure of it, and then saw that the lebesgue measure was $2i$ which is finite as $i<\infty$ , but in this case I dont know how to take elements of these sets?