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: 


1. Is $G$ of Haar measure zero in the compact group $\prod\_NG/N$?
2. 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 $G/N$?



Measure of subsets of profinite groups
1) The measure of a closed subgroup $H$ of a profinite group $G$ is $\frac{1}{\vert G:H \vert}$. So $G$ has measure zero in $\prod G/N$ if and only if it has infinite index. This way you should be able to show that $G$ always has measure zero in $\prod G/N$.


2) As Yves mentioned, you always have the inequality $\mu(S) \le \inf \frac{\vert NS/N\vert}{\vert G/N \vert}$ for any measurable subset $S$. In fact the right-hand side is the measure of the closure of $S$. If $S$ is closed it is an equality. If $G$ is the profinite completion of an abstract countable group $\Gamma$, then $\mu(\Gamma)=0$, but the closure of $\Gamma$ is $G$.


