Presentations

Proactive engagement and participation in your own learning is at the core of this course. Each lecture, the group presenter will explain the designated material (see prompts) to the other members of the group (who will have also read the material, prepared questions, and identified points of discussion ahead of the class meeting). In order to give you an idea of how the presentations should be prepared I am sharing a few videos that I made using Zoom and an iPad with pencil (Zooms allows you to share your iPad/Tablet screen so you can do this in real time). Alternatively you can consider using beamer for LaTex to prepare slides that you can share on zoom while you give your presentation. Finally, you can also prepare hand-written notes that you scan or photograph and share on zoom while you give your explanations. It is important that you expand the concise discussions in the textbook to contain additional details and thorough explanations. If you encounter a difficulty, you can either discuss it with the TA or the instructor ahead of your presentation (if it is a serious stumbling block) or within your group during the presentation (if it is a minor issue).


1. Motivation

This is a brief introduction to the subject with some motivation.

2. What should a measure be?

Here I discuss what properties a measure should have based on the intuition of length/area/volume.


3. The Issue of Measurability

Here we show that it is impossible to find a measure which is defined on the power set $2^\mathbb{R}$
of the reals and has the three properties we would like it to have and discussed in the previous video.

4. The domain of Definition of Measures

The concept of $\sigma$-algebra, which is a (non empty) collection of subsets of a set $X\neq\emptyset$
with certain closure properties, is introduced. It will play the role of domain of definition for measures.