At this point I see a blog post written by a well respected mathematician whom I feel comfortable trusting and it's being brushed aside by I don't know who.
The Amazon UK reviews of Chung's book lead me to A Probability Path by Sidney Resnick which appears to be aimed at non-mathematicians. I have invested (speaking as a renegade physicist lacking a systematic exploration of measure theory).
Some googling around suggests his students are quite pleased with him. He has, according to Wikipedia's intro for him, the undisputed king of math blogs. Both of these point to him being at least a good or above average educator.
I learned from a star student of E. Cinlar, long at Princeton. We used Royden, Rudin, Neveu, and Chung, and there were some nice topics in the course not in any of those texts, e.g., the Lindeberg-Feller version of the central limit theorem, a really nice, astounding, result on an envelope for Brownian motion, more on ergodic theory, some on additive processes, and more. Super nice course.