In a stochastic processes class, we're studying a theorem which required that a random variable $T$ have finite mean. The notes presented a counterexample where a R.V. $T$ was such that $P(T<\infty) = 1$ but for which the theorem was not valid. It then went on to explain that $E[T] = \infty$.
This could be an elementary question but if the random variable is less than infinity with probability 1, how can the mean be infinite? Both intuitive answers and math are welcome, thank you!