I'm currently writing a Bachelors thesis based on the following paper:
Douglas, R., & Pearcy, C. (1970). On the spectral theorem for normal operators. Mathematical Proceedings of the Cambridge Philosophical Society, 68(2), 393-400. doi:10.1017/S030500410004620X
In particular, I'm trying to fill in some of the "gaps" in the proposed proof of the spectral theorem. I've been able to get past most of it. However, I'm stuck on trying to prove the countable additivity of the spectral measure proposed in the actual proof of the theorem. The authors claim that it is just an easy manipulation of sets of the first category. There are two (incomplete) arguments that I have right now and I'm hoping that someone can help me in turning either of them into a proof. I will, of course, use the notation of the paper. Both arguments start out in the following way:
Let $(M_n)_{n \in \mathbb{N}}$ be a disjoint sequence of Borel sets and let $M$ denote the union of this sequence. For each $n \in \mathbb{N}$, there exists a unique compact & open set $K_n \subseteq \widehat{V}_T$, where $\widehat{V}_T$ is the abelian von Neumann algebra generated by the bounded normal operator $T$, such that $\widehat{T}^{-1}(M_n) \Delta K_n$ is meagre. Over here, $\widehat{T}$ is the Gelfand transform of $T$. At this point, this is where the arguments differ.
Argument 1: Observe that $\left(\bigcup_{n \in \mathbb{N}} \widehat{T}^{-1}(M_n) \right) \Delta \left(\bigcup_{n \in \mathbb{N}} K_n \right) = \widehat{T}^{-1}\left(\bigcup_{n \in \mathbb{N}} M_n \right) \Delta \left(\bigcup_{n \in \mathbb{N}} K_n \right)$ is a meagre set. If $\mu$ is the proposed spectral measure, then we have that: $$\widehat{\sum_{n \in \mathbb{N}} \mu(M_n)} = \sum_{n \in \mathbb{N}} \widehat{\mu(M_n)} = \sum_{n \in \mathbb{N}} 1_{K_n} = 1_{\bigcup_{n \in \mathbb{N}} K_n}$$ The argument will be complete if I can show that $\bigcup_{n \in \mathbb{N}} K_n$ is compact and open. Indeed, showing that it is open is not hard because each $K_n$ is an open set. Showing compactness is the problem and I've been trying to look for resources on Stonian spaces that would help with that (because $\widehat{V}_T$ is a Stonian space). But that's where I get stuck in this argument.
Argument 2: It follows that for each $n \in \mathbb{N}$, the set $\left(\bigcup_{k=1}^{n} \widehat{T}^{-1}(M_k) \right) \Delta \left(\bigcup_{k=1}^{n} K_k \right) = \widehat{T}^{-1}\left(\bigcup_{k=1}^{n} M_k \right) \Delta \left(\bigcup_{k=1}^{n} K_k \right)$ is meagre. Define the sequence $S_n := \sum_{k=1}^{n} \mu(M_k)$ for each $n \in \mathbb{N}$. Applying the Gelfand representation to this, we get: $$\forall n \in \mathbb{N}: \widehat{S_n} = \sum_{k=1}^{n} \widehat{\mu(M_k)} = \sum_{k=1}^{n} 1_{K_k} = 1_{\bigcup_{k=1}^{n} K_k}$$ Since $\bigcup_{k=1}^{n} K_k$ is compact and open for every $n \in \mathbb{N}$, it follows that $1_{\bigcup_{k=1}^{n} K_k} = \widehat{\mu\left(\bigcup_{k=1}^{n} M_k \right)}$ for each $n \in \mathbb{N}$. This tells us that that $\mu\left(\bigcup_{k=1}^{n} M_k \right) =S_n$ for each $n \in \mathbb{N}$; that is, finite additivity holds. Let us observe that two facts about the sequence $(S_n)_{n \in \mathbb{N}}$ hold as follows:
The sequence is increasing in the order on $\mathcal{B}(H)$; this is a consequence of the fact that $\mu$ is projection-valued and that orthogonal projections are positive operators.
The sequence is bounded because $S_n = \mu\left(\bigcup_{k=1}^{n} M_k \right) \leq \mu(\sigma_{\mathcal{B}(H)}(T)) = \text{Id}_H$ for every $n \in \mathbb{N}$.
It follows that $S_n$ converges to some operator $T$ in the strong operator topology. It should be the case that $T = \mu(M)$ and, certainly, $T \prec \mu(M)$ but I don't really have an idea of how to show that $\mu(M) \prec T$(over here, $\prec$ refers to the operator order). If I could show that $\mu(M)$ is the least upper bound of $\{S_n: n \in \mathbb{N}\}$, then I would be done because $T$ is the least upper bound of this set as well (by the monotone convergence result I'm using to conclude that $S_n \to T$ in the strong operator topology). That's where I get stuck in this argument.
So, I think that argument 1 is bound to fail anyways, just because there's a sneaky interchange of the infinite sum and the Gelfand representation on $\widehat{V}_T$ which I've not justified. Since the sum converges in the strong operator topology and the Gelfand representation is (probably) not continuous in that topology, that argument is probably doomed for failure. Nevertheless, it would be nice if it wasn't doomed for failure or if it could be fixed somehow.
On the other hand, I do think that argument 2 could probably work but I just don't really know how to finish it. If anyone has an idea for how to do this, then that would be great. I'm also curious about the argument that the authors proposed; again, they claim that it's just an easy manipulation of sets of the first category. If someone can provide that argument, then it would be quite nice.
