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I'm not too familiar with the haar measure, but I met a problem now about integration related to haar measure. Suppose I have a matrix of form $\left( \begin{matrix} U_1& 0\\ 0& U_2\\ \end{matrix} \right) $ where $U_1\in U(d_1)$ and $U_2\in U(d_2)$. If the haar measure for $U_1$ is $\mu_1$ and haar measure for $U_2$ is $\mu_2$, can I do integral over $\left( \begin{matrix} U_1& 0\\ 0& U_2\\ \end{matrix} \right) $ in form like $\int{\cdot d\mu _1d\mu _2}$? I feel direct sum in this form is independent, so I feel it should be right?

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  • $\begingroup$ Can you make your question more precise? I don't quite understand what you want to prove. $\endgroup$
    – J. De Ro
    Commented Sep 24, 2022 at 19:49

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