I am confused about the proof for the existence of Haar measure, Cartan gives a proof which avoids using the Axiom of Choice by using filter.
My question is, I try to finish the proof by using the net. I need to build a net in this way.
For all $U$, we can define $\varphi_{U}$ such that $supp(\varphi_{U})\subset U$ by Urysohn Lemma, But for every $U(\ni e)$, we have a lot of $\varphi_{U}$ such that $supp(\varphi_{U})\subset U$. Then to build a net, we actually need that for all $U$, choose a $\varphi_{U}$ From $S_U=\{\varphi_{U}|supp(\varphi_{U})\in U \}$, we can get a net by considering the union of $\varphi_U$ for all $U(\ni e)$. Do we need the axiom of choice here? we just get a choice which is not made in a uniform manner, so I guess we need the axiom of choice here? Or we just give a way to choose every element in every set? I am not sure about that.
Any help would be appreciated.