Thanks Charles and some measure theory

Dear Sumant,

Its Tuesday evening around 16:30 hrs and I am sitting on my couch at home. The shades on my huge door are turned inwards and there is no other light but the reflected sunlight that has lit up the room enough that I can type without turning on one another lamp. I am writing this because I promised that I will write everyday. Yesterday I got my car fixed and I took time to walk all the way to Tom’s place to pick it up. It was nice sunshine and walking was great. After that I went grocery shopping and then to Charles’s House to pick up the last years 501 question paper. I just asked him once and I am very thankful that he not only remembered that but sent me a text message after he found it, so that I can pick it up. Thank you Charles I really appreciate that gesture. The time has changed by 1 hr, so when I woke up today it was already 11:30 and I had a missed call from Mary. She left me a voice message regarding going to winery tomorrow. Its nice weather outside and I am looking forward to that. One thing which I still have to work on this break is getting up early in the morning. So far I am able to do things which I planned like doing Yoga everyday, updating my blog everyday, reading books etc but getting up early has still escaped my doing. The plan for today is to go and do meditation for a while, finish some proofs on 501 and do at least 2 more models of Combinatorics problem. There are few 501 proof I want to get under my belt today. The one is about the extension of concept of measure to extended real numbers. When we define measure we start with 4 goals in mind. The first is if there is a collection of sets then we should be able to define measure on any subset of that algebra. The 2^nd is called Countable Additive i.e. if we have pairwise disjoint sets then the measure of the union of sets is just summation of measure of individual sets. The third is called translation invariance. I believe it means the measure remains the same no matter where we translate that interval to i.e at all the points the measure is kind of uniform. The fourth, I think it is 2^nd or 3^rd one which states that the measure of an interval is the difference of the end points. One of my favorite theorem in measure theory is that any “Open Set” can be written as a countable union of pairwise disjoint intervals. It has a 5 step proof. First we prove that there exists an open interval which is subset of the “Open Set”. Then we prove that the union of such open intervals is equal to the “Open Set”. The fourth step is showing that if the “Intersection” of two such interval is “Not Empty” than the two intervals are “Same”. The last step is showing that there exists countable number of such open intervals because each interval contains unique rational number and since rational numbers are countable therefore the intervals are countable.

I have more to say about being finite and countable in this course. A set is Finite if it is either Empty or a range of a Finite Sequence. Similarly a set is Countable if it is either Empty or an range of a Sequence. Note when we say here sequence we automatically assume that it has infinite number of terms.

The above definition leads to few more results that can be readily derived. The very first one is “Every subset of countable set has countable number of elements”

There is another cool result Roy showed me and it states that for any countable union of collection of sets there exists an equal countable union of pairwise disjoint sets and the proof involves a standard argument which I didn’t know before. It’s easy to see once you have seen it as Roy showed me.

I have to end here

Best Regards

Sumant Sumant