How much do universities care about stuff like this? Will they really pay close attention to what you COULD have studied (e.g. measure theory)? How would they even know that a courses titled just "real analysis" does not include any measure theory? Maybe some universities have it in theirs?
Don't stress yourself out over measure theory. Real analysis without measure theory should be fine. You can teach yourself the measure theory you need as you go along -- it's not a difficult topic. Albeit it is a boring one, with the ratio of definitions to interesting theorems abysmally poor. Pay more attention to the motivation: Why is measure-theoretic probability necessary? Why is classical probability -- as expounded in the books by Feller -- not sufficient?
What's your opinion on Applied Real Analysis vs. Real Analysis? (Applied Real Analysis is considered Applied Math at my university, whereas Real Analysis is Pure Math).
Pure:
Normed and metric spaces, open sets, continuous mappings, sequence and function spaces, completeness, contraction mappings, compactness of metric spaces, finite-dimensional normed spaces, Arzela-Ascoli theorem, existence of solutions of differential equations, Stone-Weierstrass theorem.