Probability density function

Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$ and $\sup \mathbb{T} = \infty$. A function $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is called a probability density function if $f(t) \geq 0$ for all $t \in \mathbb{T}$ and $$\displaystyle\int_0^{\infty} f(t) \Delta t = 1.$$