Suppose $ f \in L^{1}(\mathbb{R}) $ satisfies $ f*f=f $. Show that $ f=0 $.
$ \hat{f} = \widehat{f*f} = \hat{f}^2 $ by problem 2.
Now $ (\hat{f}(\xi))(\hat{f}(\xi)-1)=0 $ so $ \hat{f} = \chi_{A} $ for some set $ A $
But problem 5 gives $ \hat{f} $ is continuous and the limit is zero, hence $ \hat{f}\equiv 0 $
Applying an inverse fourier transfom gives $ f = 0 $ a.e.
$ f = f*f = \int_{\mathbb{R}} f(x-y)f(y)dy = 0 $ because the integral of something that is zero a.e. is zero.
~Ben Bartle
