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.