The following is motivated by an attempt to understand the Spectral Theorem for Bounded operators on a none separable Hilbert space.
One version of the theorem states that for a bounded (say normal) operator on a Hilbert space $A \in \mathcal{L(H)}$, there exists an index set $I$ such that there is an isomorphism $$Q:\mathcal{H} \to \bigoplus_{i \in I} L^2 (S,\mu_i)$$ Where $S\in \mathbb{C}$ is the spectrum of $A$, a compact set, with a probability measure $\mu_i$, such that for $(f_i(z))_{i \in I}$ a sequence of functions in this direct sum, we have $$QAQ^{-1}(f_i(z))_{i \in I}=(zf_i(z))_{i \in I}$$ This is a remarkable result in it's own right. But one may want a single $L^2$ space to realize this direct sum. In the separable case, one can look certain direct sum of disjoint copies of $S$ with the measures $\mu_i$, $(X, \mu) = \bigoplus_{i=1}^{\infty}(S,\mu_i)$, where $\mu = \sum_{i=1}^\infty \frac{1}{2^i} \mu_i$. Then $L^2(X, \mu)$ realizes $A$ as a multiplication operator with some $F \in L^\infty(X,\mu)$. How can one do the same thing with a none separable $\mathcal{H}$? I.e. how can we realize it as a multiplication operator with a bounded function given this direct sum decomposition?. any reference or answer is appreciated.