I deal with renewal processes and do not understand the concept of the measure of a renewal process. Our professor has defined the measure of a renewal process as follows:
$$U (t) = 1_{t \geq 0} + E[N_t] $$
$N(t)$ is defined as the number of renewals untill time $t$
However, a measure should fulfil the characteristic that the measure of the empty set is zero. As we defined $U (t)$, it’s always at least one, so it can’t be a measure. Or do I have a mistake in my thinking?
I need to understand this definition to deal with renewal equations. Renewal measure is a kind of prelimeries in the notes of my Professor.
I would be very grateful if someone could explain to me this measure of renewal processes. Maybe by way of an example?
$U(t)$ is a non-decreasing right continuous function and basic measure theory tells us that there is a measure $\mu$ such that $\mu ([0,t])=U(t)$. It is this $\mu$ which is called the renewal measure.