The actual requirement for measure theory is overblown. As long as you've taken single and multivariable calculus, you can study continuous probability without any problems and without even knowing what measure theory is.
Agreed, not knowing measure theory never stopped me from computing a conditional expectation. Some courses and books overemphasize rigor in probability and, while it obviously has its place, I've seen newcomers to the field become obsessed with doing everything via measure theory. Further to your point, volume two of Feller is pretty light on measure theory IIRC.
