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.