In the first year we discussed the basics – Linear Algebra, Ordinary Differential Equations, Real Analysis and Probability. In the second year we built on those basics, studying Metric Spaces, the Riemann Integral, Group Theory and calculus on Vector Spaces.
In the third year of a four-year masters-level course, especially one with an applied focus that will be of interest to quants, we need to begin thinking about more abstract concepts that will prepare us for study of Stochastic Calculus, Probabilistic Machine Learning and Bayesian Econometrics.
With that in mind it is essential that we study topics such as Measure Theory and Linear Functional Analysis.
Both of these courses contain ideas that underlie Probability Theory, Time Series Analysis and some aspects of Machine Learning. Measure Theory teaches us about generalising the Riemann Integral to the Lebesgue Integral, while Linear Functional Analysis discusses function spaces, many of which are necessary for solutions to certain Partial Differential Equations.