Exercise 8, § 3

We identify the group $\mathbf{R}$ and its dual group by the mapping $(x, y)\mapsto$ exp($i\pi xy$). Let $\alpha \in \mathbf{R}_+$.

a) There exists a positive measure on $\mathbf{R}$ whose Fourier transform is the function $f_{\alpha}(x) =$ exp($-|x|^{\alpha}$) if and only if $\alpha \in ]0,2]$. When this is so, this measure is denoted by $\mu_{\alpha}$; it is unique; one says that it is the stable measure with parameter $\alpha$.

b) The measure $\mu_{\alpha}$ has total mass 1; the identity function $f$ on $\mathbf{R}$ is integrable with respect to $\mu_{\alpha}$ if and only if $\alpha = 2$.