Analysis – MathRants

Recently, I studied again the so called Dedekind cuts. They are a refined idea to explain how to construct real numbers out of rational numbers.

The whole problem comes from the fact that the set of rational numbers $\mathbb{Q}$ (made of the numbers that can be defined by a ratio of two integers, such as $m/n$ ) has holes. For example the number $\sqrt{2}$ doesn’t fit in $\mathbb{Q}$ .

If you don’t know why this is true, I think that you should have a look; it’s very interesting indeed.

But let’s go back to the subject of this post: Dedekind’s really smart idea.

He thought that the best way to define real number is by… not defining them! Let me explain.

For Dedekind, a cut is the division of the set of rational numbers into two disjoint, non empty and ordered sets. These two sets are just around the new number you want to define: in other words your number is the hole in between the two. Then, he defines the set of real numbers $\mathbb{R}$ as the set of all cuts.

Isn’t that smart? Yes, of course.

Unfortunately, when I read about it for the first time I felt that this idea didn’t really help my intuition:

Definition. A cut in $\mathbb{Q}$ is a pair of subsets A, B of $\mathbb{Q}$ such that:

(a) $A \cup B = \mathbb{Q}$ , $A \neq \emptyset$ , $B \neq \emptyset$ , $A \cap B = \emptyset$ .
(b) If $a \in A$ and $b \in B$ then $a < b$ .
(c) $A$ contains no largest element.

Definition. A real number is a cut in $\mathbb{Q}$ .

I closed the book and went for a coffee. After few hours thinking about it, something clicked. I had my EUREKA moment and a metaphor came to my mind.

I imagined Dedekind going to a bookshop in order to buy some books. With a detailed list of the books he wants to buy, he enters the bookshop and starts searching for them.

He heads to the right shelf and start browsing the books (that are alphabetically ordered). When he arrives at the position in the shelf where the book he is looking for should be, he finds a.. hole. Ouch, the book is not there.

Then he continues on his list and starts searching for the second book. And… this second book is also missing! Another empty slot on the shelf.

As you may know, Dedekind is a well motivated person and so he continues on his list for the whole afternoon. Imagine his disappointment when he discovers that NONE of the books he wanted to buy are available in the bookshop.

Dedekind is sad but after a few seconds (and not after a few hours like me thinking about his cuts..) he realises something: for him, it is perfectly fine to collect just the empty slots of the books he wanted to buy. After all, they are uniquely determined by the alphabetically ordered books in the shelf around each empty slot!

So he goes towards the exit and tells the cashier: “Sir, I’d like to buy all the books that are around the empty slots of the books I didn’t find. How much is it? Can you deliver them at my place by preserving the current ordering?”

The guy at the desk doesn’t really understand his questions and just answers: “Five bucks, please”.

For what it’s worth, I know now that the set of real numbers is like a large (infinite) collection of.. missing books (and only costs five bucks)!

Yesterday, I reviewed some principles behind the integration theory by Lebesgue.

In a few words: instead of slicing the domain of a function in small equal parts, Lebesgue slices the range of the function. This produces parts that are not of the same size (like the dx in Riemann) and makes possible to integrate functions that are not Riemann-integrable (for example for discontinuity).

Using this approach, the integral of a function should be the sum the elementary area contained in the thin horizontal strip between y = t and y = t − dt. This elementary area is just:

$\mu (\{x | f(x) > t\})dt.$

If we let $f^*(t) = \mu (\{x | f(x) > t\})$ then we can define the Lebesgue integral of a function $f$ as:

$\int f d\mu = \int_0^\infty f^* (t) dt$ , where the integral on the right is an ordinary improper Riemann integral.

In order to do this, Lebesgue needs a measure to compute how big is a set. This measure is the same as area in 2D and as volume in 3D but generalises in multiple dimensions.

In case a function can be integrated both with Riemann and Lebesgue integral, then the value of the two integrals is the same.

Charles Pugh, in his book Real mathematical analysis, tells a funny story about counting money on a table in the Riemann and in the Lebesgue way. The first would collect all pieces and would sum them up. The second, instead, would organise first the money in groups of similar value. It’s a useful metaphor.

I think that the following image from Wikipedia is indeed meaningful…

Category: Analysis

Dedekind at the bookshop

Lebesgue integrals