Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.

Suppose $\mu_H(H_1H_2)>0$, where $H_1H_2:=\{h_1h_2|\ h_1\in H_1, h_2\in H_2\}$.

My question is as follows : For two closed subsets $A_1\subset H_1$ and $A_2\subset H_2$ with $\mu_{H_1}(A_1)>0$ and $\mu_{H_2}(A_2)>0$, is it the case that $\mu_{H}(A_1A_2)>0$, where $A_1A_2:=\{a_1a_2|\ a_1\in A_1,a_2\in A_2\}$?