How Do We Show Integral Unity of Transformed Functions?

[hnh]

Hello, I tried this in analysis but maybe it is a more topological question. If given a function f on R such that \int_R f(x)dx = 1 and is decreasing and 1-lipschitz, show that
the function g(y) = min{x,f(x)} where y = x-f(x) and x>=0, also satisfies \int_Y g(y)dy=1.

I really would appreciate any observations. The transformation of the Domain is 1-1 right? so what else
is required to have integral unity of g also? I have some other info that may or may not help: f(x) = g(y) -1/2y +1/2|y|, x=g(y)+1/2y +1/2|y|.
Thank you

 

[Pere Callahan]

Hello, I tried this in analysis but maybe it is a more topological question. If given a function f on R such that \int_R f(x)dx = 1 and is decreasing and 1-lipschitz

I doubt that such a function exists.

 

[hnh]

I refer you to the bibliography item in my thesis:

1. B.F. Logan and L.A. Shepp. A Variational Problem for Random Young Tableaux. Ad-
vances in Mathematics, 26, 1977, 206-222.

The problem is solved now using lebesgue measure...