Overview
In standard survial analysis, the hazard (intensity) function is a function time alone. In most applications, it is thought that there are other factors that influence the probability of an event in any time period. These other factors imply that the hazard function is itself a random function.
Hazard Function Measure Theoretic Definition
To says the the hazard function depends on random factors in addition to time, is to say that the function can be calculated from the information availabel at time {% t %}. In the context of Measure Theory , this means that the hazard function is the probability of an event in a small period of time, conditioned on the time {% t %} filtration
{% P(\tau \leq t+\Delta t | \mathcal{F}_t) \approx \mathbb{I}_{\{ \tau>t \}} \lambda(t) \Delta t %}