Let $G$ be an infinite profinite group, so $$G=\lim_{\longleftarrow}G/N$$ where $N$ runs through the open normal subgroups. I have two questions:

- Is $G$ of Haar measure zero in the compact group $\prod_NG/N$?
- What is the relation between the Haar measure of a subset $E$ of $G$ and the numbers $\frac{|EN/N|}{|G/N}$, the size of the image of $E$ in $G/N$?