I am writting the following proof and would like to know if you guys think it is good enough, or should I maybe explain it further. My concern is that I maybe should prove why the last claim I make (about the integral being different) is true.
Consider the generalization of the Poisson process with a time-dependant intensity of arrivals $\lambda_t$ such that $\lim_{T \to \infty}\int_{0}^T\lambda_tdt=\infty$. The time $t\geq 0$ at which the event occurs has c.d.f: \begin{equation*} \mathbb{P}(t\leq\tau|t>s)=1-e^{-\int_0^{\tau}\lambda_tdt} \end{equation*}
Claim: in general, $\mathbb{P}(t\leq\tau|t>s)\neq\mathbb{P}(t\leq\tau-s)$
Proof.
Using the definition of conditional probability:
\begin{align*}
\mathbb{P}(t\leq\tau|t>s)&=\frac{\mathbb{P}(s if $\lambda_t$ is not a constant for all $t$.