A Very Brief Introduction to Measure Theory and the Lebesgue Integral (Part I)

In the next few posts, I shall be discussing recent topics of study that, to me at least, have been very intruiging. In previous posts, I have talked about Hilbert spaces. I have of late been considering the mathematics necessary to formally understand in a pure mathematical sense what a Hilbert space is. This post, like the others on this site, serves as a reference of newly learned topics that are of interest (to me, at least; such a comment is subjective, of course).

The purpose of this post is two-fold: (1.) to provide an update with what I’ve been up to; (2.) introduce some interesting mathematics that have expanded my understanding to the “size” of a set as well as operations such as differentiation and integration.

Here is a quick summary of what I plan to cover in the next few posts (to a brief extent):

Elementary Sets and their Measure: Here I will discuss the concept of length and try extend length in greater dimensions to that of a measure of a set. Much of this topic will rely on geometric intuition.
Lebesgue Measure: This section will dicuss the concept of Lebesgue measure and distinguish it from the elementary measure. Also brief mention will be made of measurability of sets and functions.
General Measure: Discussion will be made of a general measure as a function as well as measurable spaces and measure spaces.
Lebesgue Integral: This topic will introduce the concept of the Lebesgue integral as compared to the Riemann integral.
$L^{p}$ and $l^{p}$ spaces: This section will discuss the concept of a norm as it relates to the spaces $L^{p}$ and $l^{p}$ , and will define each space. We will also introduce the concept of Banach spaces.
Proof that $l^{p}$ space is a Banach space.

Section 1: Elementary Sets and their Measure:

The question that we want to answer is this: Given an arbitrary set, how do we go about measuring it?

In order to understand the difficulties present in this question we must first consider what are called elementary sets and the elementary measure. Elementary sets are those sets which are intuitively easy to measure; that is, intervals, rectangles, and boxes. We now give the formal definition of an elementary set:

Definition. (Interval; Elementary Set) We define an interval to be a subset of the real line $\mathbb{R}$ which take one of the following forms:

$[a,b] := \{x\in \mathbb{R}|a\leq x \leq b\} \label{(1.1)}$ ;

$[a,b) := \{x\in \mathbb{R}|a\leq x < b\} \label{(1.2)}$ ;

$(a,b] := \{x\in \mathbb{R}|a< x\leq b\} \label{(1.3)}$ ;

$(a,b):= \{x\in \mathbb{R}|a< x< b\} \label{(1.4)}$ ,

The length of an interval $I=[a,b]$ denoted $l(I):= b-a$ . For dimensions $d\geq 2$ , we define the measure of such sets as equalling the $d$ -times Cartesian product of intervals $I_{d}$ ; that is,

$\displaystyle m(B) := \prod_{i=1}^{d}l(I_{i}); \label{(2)}$

we sometimes call sets of dimension 2 or greater as “boxes.” Thus, elementary sets are those subsets of $\mathbb{R}^{d}$ such that

$\displaystyle m(E)= \bigcup_{i=1}^{d}m(B_{i}), \label{(3)}$

where $B$ is $i$ -th $d$ -dimensional box contained in $\mathbb{R}^{d}$ .

What this definition is doing is the following: first it introduces the concept of an interval and establishes the well-understood concept of its length as being the difference between the two endpoints provided one is less than the other. The definition then generalizes the idea of a length to 2 and 3 dimensions and beyond. Note that in 2-dimensions the interval then becomes a rectangle in the plane. Thus, the measure of the length of an interval then becomes the measure of the area of a rectangle. Similarly, for $d=3$ we replace rectangles with cubes and the area with the volume. For dimensions $d>3$ , we replace cubes with boxes of $d$ -dimension. Therefore, elementary sets are those subsets of $d$ -dimensional real space that are unions of finitely-many boxes.

Section 2: Lebesgue Measure

In the last section, we discussed sets for which we can measure quite easily. Though ideally we would like to be able to measure more general sets; that is, sets that are more general than elementary sets. Therefore, we require a different way of measurement. Thus, we come to need the Lebesgue measure.

In order to introduce the Lebesgue measure we need to first introduce the concept of the outer measure, which we now define

Definition. (Outer Measure) We define the outer measure of a set $E\subset \mathbb{R}$ , denoted $m^{*}(E)$ to be

$\displaystyle m^{*}(E) = \inf\bigg\{\sum_{k=1}^{\infty}l(I_{k})|\forall k\in \mathbb{N}, I_{k} \text{ is open such that } E \subset \bigcup_{k=1}^{\infty}I_{k}\bigg\}.$

The outer measure of a set in a sense “overestimates” the size of a given set and then takes the smallest such overestimate to within a specified tolerance. Thus, it estimates the size of the given set “from the outside,” and is used in lieu of the elementary measure when we are dealing with sets that we cannot easily measure the set in a geometrically-intuitive way.

We conclude this post with the definition of the Lebesgue measure given in two forms; the first will be in terms of what we have defined so far, and the second will be defined in terms that will be covered in the next post.

Definition. (Lebesgue Measure.) We define the Lebesgue measure of the set $E$ to be a set whose measure $m(E)=m^{*}(E)$ ; that is, its measure is equal to the outer measure.

The second way of defining this is as follows:

Definition. (Lebesgue Measure V.2) The Lebesgue measure is the measure on the measureable space $(\mathbb{R},\mathcal{L})$ where $\mathcal{L}$ is the $\sigma$ -algebra of Lebesgue measurable subsets of $\mathbb{R}$ that assigns to each Lebesgue measurable set its outer measure.