# Isomorphism and integral unity

1. Dec 1, 2008

### hnh

Hello, I have a non-homework problem I have been working on for a while. Let F be a set of positive decreasing 1-lipschitz functions of integral unity and G an isomorphic (to F)set of 1-lipschitz functions. F and G are defined on R and Y respectively where y=x-f(x) in Y and g in G is precisely: g(y) =min{x,f(x)} (x in R). Further, define: g(y) = 0 for all x <= -f(0) and f(x) is 0 outside a compact set and for x <0. The question? Show that g is also of integral unity, that is:

\int_[-f(0), \infty) g(y) dy = \int_[0, \infty) f(x)dx. Thank you for any input-I just realized that the 'standard' change of variable does not apply or at least is not what I need so I am stuck.