The Matrn spectral measure on the real line is proportional to the symmetric type IV distribution in

The Matérn spectral measure on the real line is proportional to the symmetric type IV distribution in the Pearson class, which is also equivalent to the Student \(t\) family (Pearson type VII). For \(v>-1 / 2\), show that the standardized version

\[
M_{1}(d \omega)=\frac{\Gamma(v+1 / 2) d \omega}{\pi^{1 / 2}\left(1+\omega^{2}ight)^{v+1 / 2}}
\]

has positive density, but the total mass is finite only for \(v>0\).