You only need measure theory when working with something that is not easily replaceable by R^n, Z^n, or finite sets to meaningfully define integration, otherwise (in)finite sums and Riemann integration get you very far.
I am a bit rusty on my advanced probability theory, but IIRC the only thing that required* measures was defining conditional probabilities and expected values on zero-probability events.
Of course redoing that class without Lebesgue integration sounds excruciatingly painful.
* Not just to make proofs nicer and theorems more powerful
You only need measure theory when working with something that is not easily replaceable by R^n, Z^n, or finite sets to meaningfully define integration, otherwise (in)finite sums and Riemann integration get you very far.
I am a bit rusty on my advanced probability theory, but IIRC the only thing that required* measures was defining conditional probabilities and expected values on zero-probability events.
Of course redoing that class without Lebesgue integration sounds excruciatingly painful.
* Not just to make proofs nicer and theorems more powerful