I was self-studying the book on Stochastic Integration by Protter for my Phd seminar in statistics and I am stuck on theorem 31 where the author proves that the filtration of the Levy Process(containing all null sets) is right continuous.
The author claims that if $$E[\exp{\iota \sum u_j X_{S_j}\vert\mathcal{G_{t+}}}]=E[\exp{\iota \sum u_j X_{S_j}\vert\mathcal{G_{t}}}]$$ for all $(u_1,\dots,u_n)$ and $(s_1, \dots,s_n)$ we have that
$E[Z \vert \mathcal{G_{t+}}]=E[Z \vert \mathcal{G_{t}}]$ for all bounded $Z \in \sigma(\mathcal{F}_s^0)$ where $(\mathcal{F}_t^0)_{t \geq 0}$ is the natural filtration of the Levy Process $X$.
Can somebody explain why is this true and why does the proof work?
I am at a total loss and I would be grateful if someone would help me out