I wanted to learn basics of the Brownian motion, and I started with the introductory book "Continuous Time Markov Processes: An Introduction" by Ligget. I have difficulty understanding something that seems very basic. He writes a theorem about the construction of Brownian motion. Theorem 1.21 states that there exists a probability space $(\Omega,F, P)$ on which standard Brownian motion B exists. Up to this point, I could somehow follow the book, but the proof is not understandable for me at all. Should the proof be understandable for a first-semester-stochastics student? What do you suggest reading before that?
(I would also appreciate it if someone could give me some intuitive explanation that might help me understanding this proof easier)
The book Brownian Motion by Peter Mörters and Yuval Peres, goes in great detail into multiple constructions of Brownian motion. They start with a nice visual construction based on the idea that random walk approximates Brownian motion.
More references include Revuz-Yor and LeGall.