Hi !

Here is my problem:

(The questions are independent)

a/

If f is a function integrable on [0,b] and

g(x) = http://www.futura-sciences.com/cgi-b...7Bt%7D%20dt%7D

pour 0http://www.futura-sciences.com/cgi-b...tex.cgi?%5Cleqxhttp://www.futura-sciences.com/cgi-b...tex.cgi?%5Cleqb.

Prove that g is integrable on [0,b] and that

http://www.futura-sciences.com/cgi-b...28x%29%20dx%7D = http://www.futura-sciences.com/cgi-b...28t%29%20dt%7D.

b/

Let f be a fonction that is measurable, finite-valued on [0,1], and such that |f(x)-f(y)| is integrable on [0,1]x[0,1].

Prove that f(x) est intégrable sur [0,1].

For part a/, using Fubini's theorem, I managed to show g is integrable, but cannot get the equality :

http://www.futura-sciences.com/cgi-b...28x%29%20dx%7D = http://www.futura-sciences.com/cgi-b...28t%29%20dt%7D.

For part b/, i set g(x,y) = |f(x)-f(y)|, integrable, and tried to use Fubini's, but don't really see what I can do with:

http://www.futura-sciences.com/cgi-b...9%20dxdy%7D%7D.

Any ideas ?