Which of the following properties does a process with independent increments really admit?

Solution 1:

You don't need $\Sigma _n$ to prove 1. (and I dont see how you can use it to prove it), and 2. is already stated in the definition of $(X_t)_t$. Just note that as $(\mathcal{F}_t)_t$ is a filtration then if $Y$ is $\mathcal{F}_s$ measurable and $t>s$ then, as $\mathcal{F}_s\subset \mathcal{F}_t$ you have that $Y$ is also $\mathcal{F}_t$ measurable. Then it follows that $X_t-X_s$ is $\mathcal{F_t}$ measurable, and so for $r\geqslant t$ we have that $X_r-X_t$ is independent of $\sigma (X_t-X_s)\subset \mathcal{F}_t$, then 1. follows easily from here.

Similarly, as $\mathcal{F}_s\subset \mathcal{F}_t$ and $X_r-X_t$ is independent of $\mathcal{F}_t$, it follows that its also independent of $\mathcal{F}_s$, this proves your last question.