Probability for mathematicians
Semester 2, 2013
Probability theory has a right and a left hand. On the left is the rigorous foundational work using the tools of measure theory. The right hand `thinks probabilistically'.
DURRETT, page xii (after Breiman).
...measure theory, that most arid of subjects when done for its own sake, becomes amazingly more alive when used in probability, not only because it is then applied, but also because it is immensely enriched.
But what really enriches and enlivens things is that we deal with lots of sigma-algebras, not just the one sigma-algebra which is the concern of measure theory.
WILLIAMS, page xi.
Of course, intuition is much more important than knowledge of measure theory.
WILLIAMS, page xii.
♦ signifies something important, ♦♦ something very important, and ♦♦♦ the Martingale Convergence Theorem.
WILLIAMS, page xiv.