Shreve's stochastic calc for finance I and II (Baby Shreve? Feels like the nomenclature should've caught on at some point...) texts were actually designed for an audience (MFE students) who aren't familiar with measure theory. I think they're excellent at what they do.
With that said, I'd imagine the probability theorists in the room went through Durrett/Billingsley/Resnick --> Oskendal/Bjork --> Shreve+Karatzas (this one) or similar.
If you're interested, check out Karatzas & Shreve for many different factoids about Brownian Motion:
https://www.amazon.com/Brownian-Stochastic-Calculus-Graduate-Mathematics/dp/0387976558