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Academic Journal of Mathematical Sciences, 2022, 3(1); doi: 10.25236/AJMS.2022.030102.

Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost


Chenyuan Tian1, Daohong Liu2

Corresponding Author:
Chenyuan Tian

1College of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, Hubei, China

2College of Mathematics and Physics, Chengdu University of Technology, Chengdu, Sichuan, China


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.


Rolle Theorem, Radon-Nikodym Theorem, Riezs Theorem, Bochner integral

Cite This Paper

Chenyuan Tian, Daohong Liu. Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost. Academic Journal of Mathematical Sciences (2022) Vol. 3, Issue 1: 8-14. https://doi.org/10.25236/AJMS.2022.030102.


[1] Wei Y H, Wang B S. Research on the relationship between radon Nikodym theorem and conditional expectation [J]. Journal of Chongqing University of Natural Sciences, 2011, 02-24.

[2] Shi Y W, Chen W L, Feng J J. Radon Nikodym theorem in complex Loeb measure space[J]. Journal of Hengshui University, 2011,13 (01): 12-13+16.

[3] Hu B J. Comparison of two proof methods of radon Nikodym theorem[J]. Journal of Chongqing Jiaotong University, 2005 (05): 167-168.

[4] Mu Y. Radon-Nikodym theorem of generalized fuzzy measure[J]. Journal of Yunnan Minzu University (Natural Science Edition), 2008(04):315-316.

[5] Ma C X. Counterexamples from Rolle's Theorem[J]. Mathematics and Applied Mathematics, Journal of Tonghua Normal University (Natural Science Edition), 2019, 5(40):33-35.

[6] Shi J H. Mathematical Analysis Tutorial [M]. Higher Education Press, 2006.

[7] Zhuo L Q. Mathematical analysis [M]. Higher education publishing house, 2006.

[8] Rudin, Principle of mathematical analysis[M]. McGraw-Hill Education, 2004.

[9] Stein. Real analysis [M]. Springer, 2013.

[10] Stein. Functional Analysis [M]. Springer, 2013.

[11] Gerald B.Folland. Real Analysis [M]. Springer, 2007.

[12] Rudin. Real and complex analsis [M]. McGraw-Hill Education, 2004. 

[13] Douglas S.Bridges. Foundations of Real and Abstract Analysis [M]. Springer, 2007. 

[14] Zhou M Q. Real-transformation function [M]. Beijing University Press, 2008.

[15] Lin Q Y. Explain analysis lectures [M]. Beijing University Press, 2007.

[16] Wang L. Research on Real Analysis [M]. Higher Education Press, 2014.

[17] Wang L. Research in the Functional Analysis [M]. Higher Education Press, 2014.

[18] Xia D X et al. Functional analysis second tutorial [M]. Higher education publishing house, 1987.

[19] Simon stevin, A simplest example of a nonmeasurable set. [J]. The American Mathematical Monthly, 1984,91(1): 22-32.

[20] Yu X T. Banach Space Geometric Theory [M]. East China Master Press, 1986

[21] Zhao H G, He J S. Several Active Conditions of Radon-Nikodym Nature [J]. Journal of Zhejiang Normal University, 1992.