Prove directly that, for each (t geq 0), [mathbb{E}left(e^{-z T(t)} ight)=int_{0}^{infty} e^{-z s} f_{T(t)}(s) d s=e^{-t z^{1

Prove directly that, for each \(t \geq 0\),

\[\mathbb{E}\left(e^{-z T(t)}\right)=\int_{0}^{\infty} e^{-z s} f_{T(t)}(s) d s=e^{-t z^{1 / 2}}\]

where \((T(t), t \geq 0)\) is the Lévy subordinator.