Finding the expectation value of a mixed function of both momenta and coordinates

I’m reading Merzbacher’s Quantum mechanics. In Chapter 3, section 2, he tackles this question. For a function solely of coordinates, say $f(mathbf r)$, he says that the expectation value is given by
$$langle f(mathbf r)rangle = intpsi^*(mathbf r, t); f(mathbf r);psi(mathbf r, t);d^3r, $$
and this makes sense to me.

But for a function like $f(x, k_y, z)$, he first defines
$$chi(x, k_y, z, t):=frac{1}{sqrt{2pi}}intpsi(mathbf{r}, t); e^{-ik_yy}; dy,$$
and says that $|chi(x, k_y, y, t)|^2$ "can easily be seen to be the probability density of finding the particle to have coordinates $x$ and $z$, but indeterminate $y$, whereas the $y$-component of it’s momentum [wavenumber]$^1$ is $p_y$ [$k_y$]."

I do observe that
$$int |chi(x ,k_y, z)|^2;dx;dk_y;dz = 1,$$
but even so, I fail to see that it should be the probability density that Merzbacher quotes.

Question: Is this—that $|chi|^2$ is the required probability—an assumption (a postulate)? Or does it, as Merzbacher seem to claim, follow from some probability considerations?

---

$^1$ Merzbacher does this for momentum, $p_y$, but I find it better to talk of $k_y$. $p_y=hbar k_y$ switches back and forth between these equivalent formulations.