Originally Posted by

**matt33** a/

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

$\displaystyle g(x) = \int_x^b\frac{f(t)}tdt$

pour $\displaystyle 0\leqslant x\leqslant b$.

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

$\displaystyle \int_0^bg(x)\,dx = \int_0^bf(t)\,dt$.

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 :

$\displaystyle \int_0^bg(x)\,dx = \int_0^bf(t)\,dt$.