Is the Lebesgue measure on [0,1]² a unique product measure on [0,1]² which is transformed again into a product measure on [0,1]² by the mapping ψ(x,y) = (x,(x+y)mod 1))? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables X and Y constancy of the conditional expectations of X+Y-I(X+Y > 1) and its square given X identifies uniform distribution either absolutely continuous or discrete. No assumptions are imposed on the supports of the distributions of X and Y.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-2-9,
author = {Joanna Chachulska},
title = {A Characterization of Uniform Distribution},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {53},
year = {2005},
pages = {207-220},
zbl = {1105.60013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-2-9}
}
Joanna Chachulska. A Characterization of Uniform Distribution. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) pp. 207-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba53-2-9/