Showing the lebesgue measure is sigma finite on sets 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? 

Well $(-i,i)$ is open, so what you did is right: $\mathbb R = \bigcup_{i \in \mathbb N} (-i,i)$.
Similarly, $\mathbb R = \bigcup_{i \in \mathbb Z} [i, i+ 1]$. 
If this is not what you asked, let me know and I'll adapt my answer.
