Can we exchange integral and limit?

Question

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Haar measure. We take an orthonormal basis {$e_i$} of $L^2(X)$.

If $f$ $\in$$L^2(X)$ with f$=\sum c_ie_i$, is $\int fd\mu =\sum\int c_ie_id\mu$ correct? If so, please tell me the proof.

Answer 1

If $X$ is compact and $\mu$ is the normalized Haar measure then this is true. If $s_n$ is the n-th partial sun of the series $\sum c_ie_i$ then $s_n \to f$ in $L^{2}$, hence aslo in $L^{1}$. This implies $\int s_n \, d\mu \to \int f \, d\mu$ which is exactly your asserion.