Lecture 02: Sigma-fields and Measurable spaces

In this lecture, we discuss the notion of σ-fields and measurable spaces. Basic properties and some examples of σ-fields are considered.

Self-assessment / Activity questions:
When do you call some collection of subsets a σ-field? Is it, by definition, non-empty?
What is a measurable space?

[RahulMadhavan]

@18:50 Question: Why can we construct 2^2^n sets?
Take each of the n sets {A_1, \dots, A_n}.
Each of the n sets may or may not be there in the result in 2 ways. So assign 0 and 1 to each of the sets depending on whether it is there or not in the output set. Thus there are 2^n distinct regions of the sample space formed (at most) by these n sets.
For example: with 3 sets {A, B, C}, the 8 disjoint partitions are: {A^cB^cC^c, A^cB^cC, A^cBC^c, A^cBC, AB^cC^c, AB^cC, ABC^c, ABC}

Now that you have 2^n distinct sets, you can choose to have any number of them as a single set. In the above example: ABC^c ∪ABC is one such set. Similarly, one can choose any combination of these 2^n sets in 2^2^n ways

[AbhishekSingh-beyv]

This professor is literally reciting his HANDWRITTEN notes! It is 2021 already. He did not even care to TeX the notes. Standing ovation for the professor. A genuine question to IIT Kanpur, is this even professional? You let this man represent IIT Kanpur on NPTEL? And now on YouTube? Okay 😂