(From the book 'Lévy processes and infinitely divisble distributions, Ken-Iti Sato, 1999', page 59/60)\
My question:
Where exactly has Lemma 11.2 been used in the proof of lemma 11.3?
Lemma 11.2: $\Omega'_2 \subset \Omega_2$ with
- $\Omega_2 = \{\omega: \lim\limits_{s \in \mathbb{Q}, s \downarrow t} X_s(\omega) \, \, \text{exists in} \, \, \mathbb{R}^d \, \, \text{for every} \, \, t \geq 0 \, \, \text{and} \, \, \lim\limits_{s \in \mathbb{Q}, s \uparrow t} X_s(\omega) \, \, \text{exists in} \, \, \mathbb{R}^d \, \, \text{for every} \, \, t > 0\, \, \}$
- $\Omega'_2 = \bigcap\limits_{N = 1}^\infty \bigcap\limits_{k = 1}^\infty A_{N, k}$ with
- $A_{N, k} = \{\omega: X_t(\omega)\, \, \text{does not have} \, \, \tfrac{1}{k}-\text{oscillation infinitely often in} \, \, [0, N] \cap \mathbb{Q}\}$
- ($M \subset [0, \infty)$, $\varepsilon > 0, \omega$ fixed) $X_t(\omega)$ has $\varepsilon-$oscillation $n$ times in M, if there are $t_0 < t_1 < \dots < t_n$ in M such that $|X_{t_j}(\omega) - X_{t_{j-1}}(\omega)| > \varepsilon \, \, \text{for} \, \, j = 1, \dots, n$. $X_t(\omega)$ has $\varepsilon$-oscillation infinitely often in M, if, for every $n$, $X_t(\omega)$ has $\varepsilon-$oscillation $n$ times in M.
Lemma 11.3
If $\{X_t\}$ is stochastically continious and $P[\Omega'_2] = 1$, then there is $\{X'_t\}$ satisfying $P[X_t = X'_t] = 1$, such that $X'_t(\omega)$ is right-continuous with left limits for every $\omega$.
Proof of Lemma 11.3
Use Lemma 11.2. For $\omega \in \Omega'_2$, define $X'_t(\omega) = \lim\limits_{s \in \mathbb{Q}, s \downarrow t} X_s(\omega)$. For $\omega \notin \Omega'_2$, define $X'_t(\omega) = 0$. It follows from this definition that $X'_t(\omega)$ is right-continuous with left limits. If $s_n \in \mathbb{Q}$ and $s_n \downarrow t$, then $X_{s_n} \rightarrow X_t$ in probability, while $X_{s_n} \rightarrow X'_t$ a.s. by $P[\Omega'_2] = 1$. Hence $P[X_t = X'_t] = 1$ for $t \geq 0$.