I didn't get any answers to my previous question; so I am trying a different tack.
I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central Limit Theorem. As a next step I would like to know the basics about Brownian motion, for example to understand one-dimensional Brownian motion in $\mathbb R$, and to be able to use the concept of Wiener measure on the path space, not just the definition, but to use it to prove interesting results, such as the Central limit theorem as mentioned in my previous question.
So, what is a suitable introductory book to Brownian motion for someone familiar with basics of probability? The reference need not prove the CLT as I had asked earlier; but if it does it will be a nice addition.