Some doubt on $L^\infty$ of measure space

Question

Is it true that $\oplus L^{\infty}(X,\mu_{n})=L^{\infty}(X,\mu)$, where $\mu=\oplus 1/2^n(\mu_{n})$, for any sigma finite class of measures?Then can you please explain how the spectral representations $L^{\infty}(X,\mu_{n})$ of will give spectral representations of $L^{\infty}(X,\mu)$. I could not understand the construction of single measure in HAHN-HELLINGER theorem by joining of measures of each cyclic representation, how the direct sum of $\oplus L^{\infty}(X,\mu_{n})$ remembers the spectral multiplicity?

Answer 1

If $\mu =\sum_n \mu_n$ then $L^{\infty } (\mu)$ is the intersection of the spaces $L^{\infty } (\mu_n)$, not their sum.