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NUS Math Module Review: MA4262 Measure and Integration

Assessment Structure:

5% Tutorial Participation (In-Class)

10% Homework (4 sets total)

25% Midterm (Online, Open Book, No Internet)

60% Final (Online, Open Book, No Internet)

This is one of the follow-up modules of MA3209, in the direction of measure theory. The module’s topics include measure and measurable functions, Lebesgue measure, integral and integrable functions, Borel measurable sets, absolutely continuous and mutually singular measures, convergence theorems of integration of non-negative functions, L^2 space and L^∞ space, Riesz representation theorem, convergence in measure and almost uniform convergence, generating measures, product measures. The main book of reference is “The elements of integration and Lebesgue measure”, R. G. Bartle.

In-class lectures were still held despite COVID-19, though all lectures were webcasted. Lecture attendance was not compulsory. Tutorial sessions were still held face-to-face, and attendance was compulsory. Tutorial sessions involve students presenting their solutions in front of the class to the rest, and presentations are accounted for in the tutorial participation portion of the assessment criteria. Prof. Wang would then proceed to give comments on the solutions (just nod in approval if the solution is good), and fill in the details if necessary. He might also provide alternative solutions if he deems it necessary. Tutorial questions are generally very manageable, with exceptions to a few questions requiring more thinking. These questions are generally built on concepts from lecture notes and serve to reinforce the concepts taught very well.

The concepts taught in this module are generally very intuitive and not hard to grasp, except the proofs in some major theorems (e.g. monotone convergence theorem) which may be quite involved and hard to appreciate. Unlike usual advanced analysis classes however, the proofs in general are not too messy and can be quite neat (e.g. proof of Riesz representation theorem). Regardless, I don’t think it is worth spending too much time understanding all the details of the proofs. Heuristics and visualisation are key in this module – understanding the concepts taught usually involve being able to visualise what the concept means, and often I am rewarded for having good visually intuitive ideas of the concepts taught when solving problems. I recommend those taking this module to spend time and energy coming up with these heuristics.

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