I have created several problem sets to practice for Oxford's final examinations. I am developing this page and will update it shortly. Some of these problems are checked by my friend, Guoxi Liu, who is also an undergrad mathematics student at Oxford. Feel free to email me if there is (I believe there is definitely a lot) any mistake. These problem sets were set to be harder than the actual Oxford exam, so please don't panic if you get stuck on them!
For other resources, please see also Cambridge Past Papers. Also, C4.1 Further Functional Analysis past papers and Cambridge Part III Exam Paper (Functional Analysis) help.
Remark. I found no functional analysis exam harder than Oxford's.
For B8.1, Probability Theory and Examples by Rick Durrett is a very good resource. Check also Cambridge Part III Exam Paper, Advanced Probability. Part II Probability and Measures may also help with the Borel-Cantelli part of the course.
For B8.2, I would recommend the classic Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor. This book is hard. Check also Cambridge Part III Exam Paper, Advanced Probability & Stochastic Calculus and Applications.
Measure Theory, Probability, and Stochastic Processes by Jean-François Le Gall is also a classic and hard book. It helps a lot with B8.1 and the first few parts of B8.2.
Remark. Some problems are from UMichigan's Math PhD Qualifying Exam, Complex Analysis papers. They are quite hard.
Complex Analysis by Elias M. Stein and Rami Shakarchi has a lot of great problems. Check also Cambridge Past Papers.
Check UMichigan's Math PhD Qualifying Exam, Real Analysis papers. They are quite hard.
Real Analysis by Elias M. Stein and Rami Shakarchi has a lot of great problems.