I am reading renewal theory from Stochastic Process by Sheldon Ross.
Proposition 3.2.2 states $E[N(t)] < \infty$ for all $0 \leq t< \infty$, where $N(t)$ is number of renewals in time t.
I understand the approach given, creating another renewal process having higher number of arrivals in t time and proving that the expected number of arrivals in time t for newly created renewal process is less than infinity.
My doubt is: In the previous page author proves that an infinite number of renewals cannot happen in finite time i.e N(t) is finite in finite time. In proposition we are given that $t< \infty$, which implies all the realizations of N(t) are finite. Doesn't that implies $E[N(t)] < \infty$?
I am unable to understand the need for a longer approach taken by the author. What am I missing?
Thanks.
A nonnegative random variable $X$ that is finite WP1, does not imply $E[X] \lt \infty$... that is why extra work is required.
Example: consider discrete $X$ that has pmf for natural numbers $\geq 2$, given by $\frac{1}{n(n-1)}$
That said, $E[X] \lt \infty$ does imply $X$ is finite WP1 (eg by Markov Inequality).