A Lévy process is defined as a stochastic process, $X = (X_t)_{t\geq 0}$, with the properties:
L1. $X_0 = 0$ a.s.
L2. $X$ has independent increments
L3. $X$ has stationary increments
L4. $X$ is continuous in probability
What is the purpose of assuming the property L4? What happens if we do not include this in the definition?
I understand that we do not want to assume a.s. continuity of paths since, as far as I understand, this would restrict us to some (non-standard) Brownian motion (like $X_t = b + \Sigma W_t$ for a standard Wiener process $W$).
Further, in the case of additive processes (processes which satisfy only L1 and L2 but not necessarily L3), then continuity in probability prevents deterministic jump times. There is a small note about this at the end of Chapter 2 of [2] and some more in [1, Sec. 37]. However, this would not be the case for Levy processes since stationarity of increments would prevent deterministic jump times (I think).
Lastly, I understand that assuming stochastic continuity implies that there exists a càdlàg modification of the process and, as such, we can work with this modification. In fact, some authors assume that $X$ is càdlàg a.s. (instead of L4). If this is the sole reason for assuming stochastic continuity, then my question becomes:
Why do we require càdlàg paths? (And why not just assume this in the first place?)
[1] Loeve, M. (1963) Probability Theory. 3 Edition, D. Van Nostrand Co., Ind., Princeton, NJ.
[2] K. Sato, Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, 1999.
