Supose $(X_t)$ is a non homogeneous Poisson process with intensity function $r$. Let $T_1$ be the first arrival time of the process and $R(t) = \int_0^t r$. Then
\begin{align*} {\bf{P}}(T_1>t) &= {\bf P}(X_t < 1)\\ &= {\bf P}(X_t = 0)\\ &= \exp(-R(t)) \end{align*}
Thus
$$R(t)=-\log({\bf P}(T_1>t))$$ and $$\lim_{t\to\infty} R(t) = \infty$$
However, for homework my professor defined $r(t) := \exp(-t)$ for which $R(t) = 1 - \exp(-t)$. We also saw some examples for which $r$ has finite support.
I guess for applications we could define $r$ as 1 after the interval of time in which we are interested, but it seems too artificial for theoretical purposes. Is this necessary? Which other properties should $r$ satisfy?
I was asked to calculate the distribution and expectation of $T_1$ for $r(t) := \exp(-t)$ . What should I write? It seems dangerous to "do probability" and use theorems with a distribution function that isn't actually a distribution, but maybe it's not too wrong?
I saw on some websites that measure theory is used for this. I know only the basics of Lebesgue measure and integration.
Thanks for the help.