Functional calculus for direct integrals Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as 
$T = \int^\oplus T_x$ for some measurable field of operators $T_x$. Suppose furthermore that every $T_x$ is self-adjoint (and so also $T$ is self-adjoint), and let $f$ be a bounded measurable function on $\mathbb R$.
Under what conditions $f(T)$ is decomposable (I guess always) and equal to the integral of the field $f(T_x)$ ?
One paper which says something about this problem is Chow, Gilfeather, "Functions of direct integrals of operators". It actually states that the only necessary condition is that $T_x$ are contractions. But unfortunately I don't understand this paper, since it doesn't state its assumptions very precisely - for example, it doesn't seem to be assumed that the operator $T$ (or operators $T_x$) is (are) normal, and so I don't what kind of functional calculus is considered.  
 A: If you want just to give a brief argument with possible references to known results, you can proceed in the following way: one picks a sequence $p_n$ of polynomials converging to $f$ in the weak-measure topology on the Borel functions; then $p_n(T)$ converges to $f(T)$ even strongly (see e.g. Helemski. Lectures and exercises on functional analysis, p. 388). As $p_n(T)$ commuted with every diagonal operator, $f(T)$ does commute as well, and therefore is decomposable (Dixmier. Les algèbres d'opérateurs dans l'espace hilbertien, Thm. II.2.5.1), say, as $\int^\oplus S_x d\nu(x)$. Now, there is a subsequence $p_{n_k}$ such that $p_{n_k}(T_x)$ converges strongly to $S_x$ $x$-almost everywhere (Dixmier, Prop. II.2.3.4), so $S_x=f(T_x)$ almost everywhere.
A: Your guess that it is always decomposable is correct.  Here is a way to see this without verifying the expected formula: Borel functional calculus keeps you inside the von Neumann algebra generated by $T$, and the set of decomposable operators on $H$ is the von Neumann algebra of operators that commute with the diagonal operators on $H$ (Kadison-Ringrose 5.2.8, Takesaki 8.16; see also K-R 14.1.10 which has no Google preview).  
(However, I don't know a reference (or have a proof) that the expected formula is correct.  I think it should follow by Fubination once the case of characteristic functions is known.)
A: I wanted to make it a comment to Jonas' answer, but the system didn't allow me (because it's too long?)
I might be forced to write down my own proof of the expected formula. What do you think about the following sketch? The statement is clear for polynomials. Then take a sequence of polynomials $p_n$ converging somehow to $f$ (How?). This should imply that $p_n(T)$ converges weakly to $f(T)$ and similarly for $p_n(T_x)$ for a.e. $x$. Now one needs to check that $f(T_x)$ is a measurable field, i.e. whether $lim_n (p_n(T_x)v_x,w_x)$ converges to a measurable function whenever $v_x, w_x$ are measurable vecotr fields. But this limit is the same as $lim_n ([p_n(T)]_xv_x,w_x) = ([f(T)]_xv_x,w_x)$, because by your argument we know that $f(T)$ is decomposable (there is some argument needed here). The expected formula then holds because of the same reasoning and uniqueness of the weak limit.
