In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove the claim in Appendix 3 of the same chapter and says that the finite action contribution to the path integral must be zero. This really confuses me.

When we perform a saddle point approximation about a finite action field configuration, we do it based on the fact that the maximum contribution to the path integral comes from configurations around this saddle point. If what Coleman claims is true, then the saddle point approximation makes no sense.

What am I missing here? It would be great if someone could clarify this.