This is a question which has plagued me for a long time.
In the path Integral, we insert$\int\mathcal{D}\Phi(x)\vert\Phi\rangle\langle\Phi\vert=\mathcal{I}$ and $\int\mathcal{D}\Pi(x)\vert\Pi\rangle\langle\Pi\vert=\mathcal{I}$
where $\hat{\phi}(x)\vert\Phi\rangle=\Phi(x)\vert\Phi\rangle$, $\hat{\pi}(x)\vert\Pi\rangle=\Pi(x)\vert\Pi\rangle$ and $\hat{\phi}(x)$,$\hat{\Pi}(x)$ are the field operator and its canonical conjugate operators.
In principle, we should and could verify these completeness relations. But it appears to be tedious work. Meanwhile, I'm not even sure about what the integration measure means. What I have so far is the field operator eigenstates expanded in creation and annihilation operators. See the answer to this problem https://physics.stackexchange.com/q/706613.
The next step should be properly defining the integration measure and calculation by brute force. The measure is continuous limit of discretization of spatial point in practice. I'm not looking to do it myself. There must have been someone who proved this rigorously. I'll appreciate it if someone can post an answer or refer me to relevant resources.
I can think of a possibly unrigorous argument: field operators are Hermitian so their eigenstates must form a complete basis of the Hilbert space. But I'm not convinced that such an argument should close the case. Since the field theory grows out of Fock space, we should be able to verify it explicitly. Moreover, a proper definition of integration measure is essential.
