why the sigma algebra generated by null set and Brownian Motion is right continuous?

Question

I mean why the generated one satisfies the definition of right continuous?

Answer 1

There is the following result, I quote from Karatzas/Shreve:

For a $d$-dimensional strong Markov Process $X=\{X_t;\mathcal{F}_t^X;t\ge 0\}$ with initial distribution $\mu$, the augmented filtration $\{\mathcal{F}_t^\mu\}$ is right-continuous.

This is Proposition 2.7.7. in Karatzas and Shreve. Since Brownian Motion has the strong Markov Property, you get your desired result. A proof of this can also be found in the book.