I have just come across direct integrals so my understanding is shaky, hence I apologize in advance if anything I say is incorrect.
Given a measure space $(X, M, \mu)$ and a collection of separable Hilbert spaces $H_x$, each with inner product $\langle \cdot, \cdot \rangle_x$ we may define the direct integral of the collection $\{H_x\}_{x \in X}$ with respect to $\mu$ to be the collection of all equivalence classes (where the equivalence relation is equality almost everywhere) of functions $f$ such that $$\|f\|_\oplus = \int_X \langle f(x), f(x)\rangle_x \,d\mu(x) < \infty.$$ We denote this space by $$\int_X^\oplus H_x\,d\mu(x).$$
The textbook I am using (Quantum Theory for Mathematicians By Brian Hall) gives two examples of this construction. The first is if $H_x = \mathbb{C}$ for all $x \in X$, then it is easy to see that the resulting direct integral is simply $L^2(X, \mu)$.
The second example is suppose $X$ is countable, $M$ is the $\sigma$-algebra of all subsets of $X$, and $\mu$ is the counting measure. However this one I am unable to see for myself. How exactly does this hold?
On a side note, why do we even care about this construction in the first place? I see it comes up often in the context of spectral theory, but what exactly motivates its existence?