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This is a separate question altogether, because now you are talking about true integration and not just a notation for the application of a linear functional. …

The measure $d^4p$ is trivially invariant under Lorentz transformations. The reason is that under the transformation $p^\mu\to p'^\mu=\Lambda^\mu_{\phantom\mu\nu}p^\nu$ the measure transforms by a Jac …

This is notation from Distribution Theory in Functional Analysis. The theory of distributions is meant to make things like the Dirac Delta rigorous. In this context, just to give you one overview, a …

_0(\phi)$ for the free action $S_0[\phi]$ is known, one may perturbatively build $d\mu(\phi)$ for the interacting action $S[\phi]=S_0[\phi]+gV[\phi]$ by really defining the interacting $Z[J]$ through integration …

View $p^2-m^2$ as a function of $p^0$ and in the end use the above result to eliminate the integration over $p^0$ to understand the relation with the left-hand side. That's a good exercise. …