Consider a measure space $(X, \mathcal{B}, \mu)$. By quotienting out its null sets, we get the measure algebra $(\tilde{\mathcal{B}}, \tilde\mu )$. By considering square-integrable functions out of $X$ into complex numbers, up to $\mu$-almost everywhere equality, we get the Hilbert space $\mathcal{L}^2(X, \mathcal{B}, \mu)$.
Thus we get three ways to study the "same" thing, that of some kind of blob, which can be cut into smaller blobs, and whose size can be measured.
In the first viewpoint, the measure theory viewpoint, we study sets of points. The style is set-theoretic, pointwise, often digging down into considerations over individual elements, like set-theoretic topology.
In the second viewpoint, the measure algebra viewpoint, we study an algebraic structure. Each algebraic element hints at the set of points it's supposed to represent, without granting access to the points. The style is algebraic, like pointless topology, or category theory.
In the third viewpoint, the Hilbert space viewpoint (or the "spectral viewpoint"), we study an analytic structure. The style is analytical, like functional analysis, complex analysis, spectral theory.
The three viewpoints are used in measure theory, ergodic theory, probability theory, etc, but I haven't seen any book explaining in a concise way how they relate to each other.
The passage from measure space to measure algebra is easy, just take quotient.
The passage from measure algebra to measure space is harder. It relies on the Loomis-Sikorski representation theorem, as detailed in this post by Terry Tao. This can be thought of as a more difficult (the difficulty comes from adding countable infinity to the mix) version of the Stone representation theorem, which allows any Boolean algebra to be represented as an algebra of sets.
The passage from measure space to Hilbert space is just the standard construction of $\mathcal{L}^2(X, \mathcal{B}, \mu)$.
The passage from measure algebra to Hilbert space is a bit harder, but it can be done. Start by defining some vectors $$\{\chi_B : B \in \tilde{\mathcal{B}}\}$$ with "pointwise multiplication", $\chi_B \cdot \chi_C = \chi_{B\wedge C}$, and addition: $\forall B, C \in \tilde{\mathcal B}$, if $B\wedge C = 0$, then $\chi_B + \chi_C = \chi_{B\vee C}$, and an inner product $$\forall B, C \in \tilde{\mathcal B}, \text{ if } B\wedge C = 0, \text{ then } \left<\chi_B, \chi_C \right> = 0$$ and also $$\forall B \in \tilde{\mathcal B}, \left<\chi_B, \chi_B \right> = \tilde\mu (B)$$ This gives an inner product space. Take the metric completion to get the Hilbert space $\mathcal L^2(\tilde{\mathcal B}, \tilde \mu)$.
The passage from Hilbert space (with pointwise multiplication) to measure space is obscure, and I cannot find any reference on that. My guess is that given an abstract Hilbert space $\mathcal{H}$ with some extra structures, we can recover a measure space $(X, \mathcal B, \mu)$, such that $\mathcal{H} = \mathcal L^2(X, \mathcal B, \mu)$, but I don't know.
Anyone knows?