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This is from Conway's operator theory book, chapter 6, exercise 3.

If $X$ is compact and $\mu$ is a positive Borel measure on $X$, then $\pi_\mu$ : $C(X) \rightarrow \mathcal{B}\left(L^2(\mu)\right)$ defined by $\pi_\mu(f)=M_f$ is a representation of $C(X)$. Consider the two positive Borel measures $\mu$ and $\nu$. Show that $\pi_\mu \oplus \pi_\nu$ is cyclic if and only if $\mu \perp \nu$.

My attempt:

Forward direction:

Suppose $\exists (f_1, f_2)\in L^2(\mu) \oplus L^2(\nu)$ such that $\{(\pi_\mu(g) f_1, \pi_\mu(g) f_2) \mid g \in C(X) \}$ is dense in $L^2(\mu) \oplus L^2(\nu)$

$\implies \{(g f_1, g f_2) \mid g \in C(X)\}$ is dense in $L^2(\mu) \oplus L^2(\nu)$.

I'm having trouble understanding the structure of $L^2(\mu) \oplus L^2(\nu)$ where $\mu \perp \nu$.

Any help is appreciated.

Vinay Deshpande
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    So you can divide $X$ into two subset $U, V$ where $\mu$ is zero on measurable subsets of $V$ and similarly for $\nu$ and $U$. Then given any $g_1\in C(U)$ and $g_2\in C(V)$ you can uses Tietze's extension theorem as the space is compact(and im assuming Hausdorff) to get a function $g$ so that $(gf_1, gf_2)$ looks like $(g_1f_1, g_2f_2)$ for practical purposes, and then you just have to show then density of ${gf\ |\ g\in C(U)}$ in $L^2(U)$. – Niranjan Kumar Jun 08 '23 at 14:59
  • Thanks to your comment the query is resolved. – Vinay Deshpande Jun 09 '23 at 07:16
  • @NiranjanKumar Can you please also help me solve the whole problem? Per your hint, I made this (https://math.stackexchange.com/q/4718594/628415) attempt. But I didn't recieve any response there. Thanks! – Vinay Deshpande Jun 21 '23 at 04:02

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