Claim: suppose that $E$ is a set of finite perimeter, and $H$ is a half space. Then $P(F\cap H)\le P(F)$. In words: restricting a Caccioppoli set to a half-space will not increase the perimeter.

My question: how to prove this?

I have a feeling it should be very simple, as in Almgren, Taylor, Wang's paper "Curvature-driven flows: a variational appraoch", it is claimed that it follows from "Stokes' theorem and Jensen's inequality". However, I'm having trouble to apply Jensen's inequality to this problem, except in trivial cases that boil down to linearity of the integral.

EDIT: Although no satisfying answer has been given as to what Jensen has to do with this theorem, a complete proof has been given. I've accepted this as an answer.