Drawing: Definition of Lebesgue Outer Measure

The Lebesgue Outer Measure, or more briefly the outer measure, of a set is given by m^*(A)=inf \sum l(I_n), where the infimum is taken over all finite or countable collections of intervals I_n (where I_n = [a,b) is bounded) such that A\subseteq\cup I_n

G De Barra

While we were thinking about how to construct examples for this definition; a friend came up with this interesting observation:

Step 1: We will collect the coverings of set A.
Step 2: We will find the summation of the lengths of the elements in each covering.
Step 3: We will put all the values what we got in Step 2 in a set.
Step 4 : We will find the inf[imum] for the set we constructed in Step 3.

I was quite excited by her observation. I thought I’ll give it a cartoon twist, to help them learn the definition better.

Visual queues cues to remember things is very useful trick students can leverage to memorize key facts.

In fact, if you have watched BBC’s Sherlock series, you might have heard a term called “Mind Palace” over and over again – that’s an extension of the use of visual queues to remember facts, but takes a lot of practice and meditation to build mind palaces. Read more about it in Wikipedia >>

Another simpler tool you can use to organize your thoughts is called Mind Maps. This is a tool you can master easily. Read more about it in Wikipedia >>

We used to use mind maps to summarize learning in class – do try it out, and get in touch with us if you need any help!

We had talked about some useful computer-aided mind mapping tools in a recent post (link).

Don’t forget to hit the like button below/share this article if you find it interesting! 🙂 If you do like this, do follow the blog for more posts like this!

This cartoon also reminds me of a very good blog I came across many years ago: Math with Bad Drawings (link). Must confess he’s been an inspiration to me! This recent post (link) there might interest – do check it out!

Recommended Books for Real Analysis

Introduction to Real Analysis by Bartle and Sherbert (4e)
Principles of Mathematical Analysis by Rudin
Real Analysis by Kumaresan
Introduction to Topology and Modern Analysis by Simmons
Topology of Metric Spaces


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