I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph [2], second chapter or alternatively his paper [1] which contains basically the same proof.

Let $\Phi\subset H\subset \Phi'$ be a rigged Hilbert space, $\Phi$ nuclear. We want to prove, that there exists a complete set of generalized eigenfunctions in $\Phi'$.

The basic idea is the following: We use the Integral formulation of the Neumann spectral theorem, i.e. every normal operator on a separable Hilbert space $H$ is unitary equivalent to a multiplication operator on a direct integral of Hilbert Spaces (which is basically an $L^2$-type complex-valued function space), the latter shall be called $\hat{H}$ and the unitary transformation $F:\varphi\rightarrow \varphi(x)$. The key element in the proof is now, that we can prove, that evaluating some $F\varphi$ at a point of the spectrum, say $\lambda$, is a continuous mapping into the complex numbers and therefore associated with some element of the dual space. Concretely, Maurin "shows" that that the mapping $F(\lambda):\varphi\rightarrow\varphi(\lambda)$ is continuous for every $\lambda$ up to a set of measure zero.

**I highly dout that this mapping is even well-defined**, as the functions in the direct integral of Hilbert spaces are only defined up to an equivalence class, i.e. they can differ on a set of measure zero.

Am I overlooking something trivial? Is there a simple way to patch this up? Is there some reference which solves this issue? I have heard that the proof given by Gelfand and Vilekin also has a similar issue. I would greatly appreciate any help, I am very stuck at the moment.

**References**

[1] Krzysztof Maurin,"Allgemeine Eigenfunktionsentwicklungen. Spektraldarstellung abstrakter Kerne. Eine Verallgemeinerung der Distributionen auf Lieschen Gruppen". (German) Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 7, 471-479 (1959), MR114133, Zbl 0106.08903.

[2] Krzysztof Maurin, *General eigenfunction expansions and unitary representations of topological groups*. (English)
Monografie Matematyczne. 48. Warszawa: PWN - Polish Scientific Publishers. 367 p. (1968), MR247377, Zbl 0185.39001.