Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.

Suppose now that I have a bounded operator $T$ on $\cal H$, about which I know that it is decomposable.

Do you know of any kind of a "formula" which will "compute" a measurable field of operators $T_x$, such that $\int T_x =T$?