[afiori]

You can do quite a lot of useful stuff with just https://en.wikipedia.org/wiki/Cumulative_distribution_functi...

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