We give an elementary proof for the case of the circle group of the theorem of O. Hatori and E. Sato, which states that every measure on a compact abelian group G can be decomposed into a sum of two measures with a natural spectrum and a discrete measure.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-1-1,
author = {Przemys\l aw Ohrysko},
title = {An elementary proof of the decomposition of measures on the circle group},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {1-4},
zbl = {1323.43001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-1-1}
}
Przemysław Ohrysko. An elementary proof of the decomposition of measures on the circle group. Colloquium Mathematicae, Tome 139 (2015) pp. 1-4. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm140-1-1/