Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost

Abstract

This paper are divided into two mains parts with different affairs to discuss. The first part are some history of how the modern strict mathmaticala analysis built up. And there’s a simple counterexample for Rolle theorem. As we all known Rolle theorem has three requirement to work out. So, there’s some counterexamples for which there’s only two requirements exist, and what is going to happen. The second part gives two counterexamples on Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Then there’s a proof for the equivalent of the Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Some recent research are given at the last, which means it’s still a vigor field nowadays.