I intend on taking a Measure-Theoretic Probability course in the fall at the level of Rick Durrett's Probability: Theory and Examples.
I am self-studying Baby Rudin chapters 1-7 this summer, and I am currently on chapter 6. I am asking the following:
If I have an excellent understanding of Baby Rudin ch.6 (on the Riemann-Stieltjes integral), is this sufficient pre-requisite knowledge for Daddy Rudin ch.1?
(Edit: I missed the part about Chapter 1 of Big Rudin. My answer is yes, study Chapter 7 of Baby Rudin. Sequences of functions figure prominently in Chapter 1. Chapters 8 and 9 might not be necessary right away, but you will need them for probability anyway. The answer below assumed you intended to go further in Big Rudin.)
You might be able to get by, because there are few hard prerequisites, but in terms of mathematical maturity, I can't imagine studying much of Big Rudin without knowing most of the material from Chapters 7-9. It's okay if you've studied it elsewhere.
Ideas about uniform convergence (Chapter 7 of Baby Rudin) come up regularly in Big Rudin. Even more complicated variations on this idea appear, so it's best to be comfortable with the basics.
Power series (Chapter 8) come up in complex analysis, obviously. I don't recall offhand if they come up before then, but this is important background knowledge for anyone.
The positive results on pointwise convergence of Fourier series in Chapter 8 of Baby Rudin are important background knowledge for the negative results in Chapter 5 of Big Rudin. It would be a bit strange to read things on Fourier series in Big Rudin without having studied this.
Differential calculus (Chapter 9) is necessary even to understand the statement of the change-of-variable formula, let alone its proof.
So the answer depends on your prior knowledge of analysis. But assuming you don't know Chapters 7-9 from elsewhere, I would say that you'd have to have some good reason to want to dive into Big Rudin first.
Even with respect to probability, all of these topics are important.