Moment generating function

Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$ and $\sup \mathbb{T}=\infty$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a probability density function. The moment generating function of $f$ is defined to be $$M_f(z) = \displaystyle\int_0^{\infty} f(t) e_z(t,0) \Delta t.$$

References

Probability theory on time scales and applications to finance and inequalities by Thomas Matthews