Present one of your many more solutions, to see where my answer is wrong.

@Colin: Do you see the answer below?

No, there are not infinitely many. The pieces must be identical, and must touch the center point.

Do you have an answer to the initial question? Is there a connection between the Haar measure of the big group and the Haar measure of a compact subgroup?

I was going to ask just that. The answer given uses compactness of the weak star topology, and the first question is still not answered: what is the connection between the Haar measure on the group and the Haar measure on one of its compact subgroups, if there is any?

Thank you. It was indeed simple, but I didn't encountered many such arguments before.

Yes, it is in the chapter on Haar Measure in Katznelson, Harmonic Analysis, but the theory presented there I couldn't find anything that would help me solve this.