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

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