# The Hahn and Jordan Decomposition Theorems

 Title: The Hahn and Jordan Decomposition Theorems Authors: Marie Cousin (marie /dot/ cousin /at/ grenoble-inp /dot/ org), Mnacho Echenim (mnacho /dot/ echenim /at/ univ-grenoble-alpes /dot/ fr) and Hervé Guiol (herve /dot/ guiol /at/ univ-grenoble-alpes /dot/ fr) Submission date: 2021-11-19 Abstract: In this work we formalize the Hahn decomposition theorem for signed measures, namely that any measure space for a signed measure can be decomposed into a positive and a negative set, where every measurable subset of the positive one has a positive measure, and every measurable subset of the negative one has a negative measure. We also formalize the Jordan decomposition theorem as a corollary, which states that the signed measure under consideration admits a unique decomposition into a difference of two positive measures, at least one of which is finite. BibTeX: @article{Hahn_Jordan_Decomposition-AFP, author = {Marie Cousin and Mnacho Echenim and Hervé Guiol}, title = {The Hahn and Jordan Decomposition Theorems}, journal = {Archive of Formal Proofs}, month = nov, year = 2021, note = {\url{https://isa-afp.org/entries/Hahn_Jordan_Decomposition.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.