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Math and science::Analysis::Tao, measure

A motivation of measure [stub]

[...] is a motivational progression though different ways of describing and measuring the volume of things in \( R^d \) space.

The standard integral

A lot of people feel an intuitive understanding when they see something like the expression \( A = \int_{a}^{b} f(x) dx \) and a figure like:

TODO: insert figure.

Maybe they just have come to forget the parts of this formulation they didn't come to appreciate on an intuitive level.

When I look at the expression I interpret \( A \) as the "area" under the graph, where my concept of "area" is an intuitive one, rather than an a concept arising from some mathematical definition.

Integration in the form of a challenge-response exchange

The following exchange starts to approach more closely what \( A \) really represents mathematically.

Imagine wise looking man approaches you as you walk down the street. His name is Darboux and wants to show you his trick. He has a special way of filling and covering things with rectangles. Specifically, he shows you a figure [TODO, show x-y function]. And he says, for this figure he has a special number, 7.214.

He challenges you: no matter what positive number \( \varepsilon > 0 \) you choose, he can do two things:

  1. He can find a set of non-overlapping rectangles (except possibly for their edges) that fit inside \( A \), and these rectangles will have a combined area (sum of \( \text{height} \times \text{width} \)) that is greater than \( A - \varepsilon \).
  2. I can find a different set of non-overlapping rectangles (except possibly for their edges) that contain \( A \), and these rectangles will have a combined area (sum of \( \text{height} \times \{width} \)) that is less than \( A - \varepsilon \).

Furthermore, he says, he doesn't need infinite rectangles, he will do it with a finite set of rectangles: for each \( \varepsilon > 0 \) you give, he can tell you the number \( N \) of rectangles he needs in order to fill or cover the function.

After he convinces you of his ability, he then challenges you! Try find another number other than 7.214 and see if you can fill and cover the function with rectangles like above. Better to not take up this challenge, for it is impossible.

This is a challenge response interpretation of the limit formulation of the classic integral.

Shapes in \( R^d \)

Darboux could only fill and cover the space under a function \( \mathbb{R} \to \mathbb{R} \). A few months later, as you are walking down the same street as before, a different wise looking man approaches you. His name is Jordan, and he too wants to show you a trick. He said that Darboux's trick was weak in comparison to his trick! Not limited to filling and covering the space under a function, Jordan says: "pick any shape in 3D or any \( \mathbb{R}^d \) Euclidean space and I will find a special number \( A \in \mathbb{R} \) and two sets of rectangles, one that covers and one that is contained within the shape. Like before, there won't be an infinite number of rectangles, they won't overlap (except possibly the edges) and the volume of the rectangles in each set will sum to the special number \( A \).

The breakdown of Jordan measure 1

Jordan measure fails to give measure to countable unions or intersections of sets that are already known to be measurable. The bullets are an example.

The breakdown of Jordan measure 2

Imagine using the Jordan measure to determine probabilities. When requested for a probability, he convert the problem into a geometric representation in \( \mathbb{R}^d \) then calculate the "volume"/ "measure" of both the space representing all possible events and the "volume" of an event in question.

However, here is a case where the Jordan measure will breakdown. Imagine a random number generator that generates real numbers between 1 and 2. What is the probability that the number generated is a rational and what is the probability that the number generated is an irrational? This can be though of as a question about the relative abundance of rationals and irrationals within \( [1, 2] \).

Jordan's approach is to calculate the "volume" of the rationals: try to find two sets of disjoint rectangles, one set of that covers the rationals and one that is contained within the rationals. But the only boxes that fit within the rationals have zero volume. For example, \( (\frac{4}{3} \) and \( \frac{4}{3}) \), so the set of boxes that are contained within the rationals cannot have a combined volume greater than zero. This would suggest that Jordan's measure is 0. However, when we investigate the outer cover, we encounter a problem. If we restrict ourselves to finite sets of boxes, we can't cover the rationals with disjoint boxes with a combined volume less that 1. [TODO: tidy up terminology of boxes and intervals and rectangles]

So we have two different numbers for our volume measurement: 0 or 1. If we relate these back to the question of probability, this means that the random number generator will either never or always produce rationals.

Lebesgue measure

The challenge-response interpretation of Jordan measure for a shape involved choosing an \( \varepsilon > 0 \) and having Jordan find a two finite sets of boxes--one set that covers the shape and one that is contained within the shape in such a way that both box covers have the same volume when calculated as the sum of the volumes of each box.

The Lebesgue measure relaxes the requirement for the box cover to be finite. Given an \( \varepsilon > 0 \), Lebesgue can find a countable set of boxes that covers a shape.

Darboux vs Riemann integral

The Darboux integral and the Riemann integral are two equivalent ways of formulating the classical integral. They are equivalent in the sense that they always have the same value (when they work), and they break down in the same circumstances. It's as if they implement the same programming interface and always produce the same output for a given input. They differ in how they define the integral.

The Darboux integral has two parts: an upper and lower integral. The upper is the infimum of volume for almost-disjoint rectangle covers of the area under a function. The lower is the supremum of volume for almost-disjoint sets of rectangles contained within the space under a function. When the supremum and infimum are equal,

or the Riemann integral)

Transition to measure any shape (Jordan measure)

Jordan measurability

The most interesting thing: what is not considered "measurable" an why?

Transition to uncountable boxes

Measurability again