A Characterization of Uniform Distribution
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 $\psi(x,y)=(x,(x+y)\bmod 1))$? 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 $Y$.