As a main reference we will use
[D] B. Driver "Analysis tools with examples"
In the last part of the course we will also use
[R] D. Romik "Probability theory"
Here are some other options:
[P] G. Pedersen "Analysis now"
[MDW] J. McDonald, N. Weiss "A course in real analysis"
[KF] A. Kolmogorov, S. Fomin "Elements of the theory of functions and functional analysis"
[C] J. Conway "A course in functional analysis"
[Du] R.M. Dudley "Real analysis and probability"
[CFS] I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai "Ergodic theory"
For the exam you need to know the material covered by Driver in sections 18.4, 20.1, 21.2, 24.1-24.3, 25.1-25.3, 28.2, 28.4, 30.1, 30.2, 30.4 and by Romik in Chapter 2 (conditional expectations). The last several lectures on ergodic theory are not part of the curriculum.