Help Me Understand Cadlag Modification of Levy Processes

[Focus]

I was reading Stochastic Integration and Differential Equations by Protter and it had a nice theorem that every Levy process in law which has continuity in probability, admits a cadlag modification. The proof is very confusing and I was wondering if anyone could help me clear it up a bit.

I wish to prove a slight modification, which is that the set
$$\{\omega : \not \exists \lim_{\mathbb{Q}\ni s\downarrow t}X_t(\omega) \quad t \geq 0 \}$$
is measurable and has a measure zero. I have that X has stationary independent increments and is continuous in probability (also X starts at 0 a.s.). Any help would be much appreciated.

 

[Valkarie]

Let's start by assuming that the set $\{\omega : \not \exists \lim_{\mathbb{Q}\ni s\downarrow t}X_t(\omega) \quad t \geq 0 \}$ is not measurable, which implies that it has a positive measure. Then, since X has stationary independent increments, we can use the strong law of large numbers to find that for some $N>0$ and $t\geq 0$, $$P\left(\frac{1}{N}\sum_{n=1}^{N}X_{t+n}-X_t>\epsilon\right) > 0$$for some $\epsilon>0$. However, this contradicts the fact that X is continuous in probability. Therefore, the set $\{\omega : \not \exists \lim_{\mathbb{Q}\ni s\downarrow t}X_t(\omega) \quad t \geq 0 \}$ must be measurable and have measure zero.