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Is real analysis a hard prerequisite for measure theoretic probability?

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You will need a solid foundation in the basics of analysis (up to Riemann integration, as you say), especially when it comes to convergence of sequences and series, especially sequences and series of functions. Also, theoretically you could learn Lebesgue integration and measure theory without having seen some things like basic topology and Riemann integration, but I doubt it would be so easy in practice.

Imo you can get away without learning measure theory and Lebesgue integration before taking a probability course, as in my experience you rehash these things in the language of probability anyway, and a lot of measure theory concepts make more sense in the context of probability (e.g sigma algebras). But you will still need to be very well-versed and comfortable with the basics.
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Maybe not a hard one, but it would definitely help. Measure theory is often perceived as very challenging and not having to struggle with the language of it would definitely make it easier.

It would also give some meaning to measure theory I guess.
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throw in Lebesgue integration and you're good to go I'd say. They will cover a lot of the main results of Lebesgue in probability, but they'll do it abstractly, i.e. under arbitrary probability space, so it helps to have seen it in the reals to develop intuition.
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I would say it is a the heart of measure theoretic probability.

I'm curious as to what others smarter than me may have to say.

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