What is the connection between the normalized Haar measure of a compact group and the normalized Haar measure of one of its compact subgroups?

I am trying to solve the following problem:

Given $G$ a compact group with normalized measure $\mu$ and $\{H_n\}$ an increasing sequence of compact subgroups of $G$ with normalized Haar measures $\mu_k$ such that $\bigcup H_n$ is dense in $G$. Prove that $\mu_k$ converges in the weak star topology to $\mu$.

[edit] The problem is indeed an exercise, as you can see from my comments, but I don't know why this is so relevant. I asked a question which could enlighten me in order to solve the given problem, and I think that the given question about Haar measures is not so trivial, since no one gave an answer until now.

extremelyrelevant, whether you realize it or not. At the very least, by pinpointing where the problem can be found, you give users of this site an idea of what you are expected to use to solve the exercise in question, which can give a useful hint of where to start, but also isespeciallyimportant when there may be completely different approaches to solving a problem. So kindly get off your high horse, thanks. $\endgroup$ – Thierry Zell May 23 '11 at 18:24questions, like you give in your first paragraph. I would be happy for you to tell me that what you're looking for is an answer to your first question, and your motivation is towards being able to solve the exercise you quote. But you do not make this clear in the post. $\endgroup$ – Theo Johnson-Freyd May 24 '11 at 0:55