Lebesgue Theory Part 1 (Set Functions)

What is a set function? In fact, what is a function? Before jumping into the deep water, let us take a moment to review the idea of a function, what it is, and how it is useful to us.

1) Definition: Given two non-empty sets ${A}$ and ${B}$ a function ${f}$ is a rule (i.e. a set of instructions) where each element of ${A}$ is “paired” with exactly one element of ${B}$ . Set ${A}$ is denoted as the domain of ${f}$ and ${B}$ is denoted as the range or codomain. We use the notation, ${f}:A\rightarrow{B}$

Now that we have cashed-out the idea of what a function is defined as, we can move toward our defining a “set function.” However, let’s take a step back and address the basics of sets and what they are. A function is perhaps one of the most important mathematical concept that exists. In fact, the essence of a function is merely, upon reduction and distillation of all the abstract fluff, an encoding of of information from one group of objects to another. This encoding process happens according to a well-defined list of rules.

I mean, do you like to cook? Have you ever baked a cake? You take one group of “stuff” and add other “stuff” in a specific fashion to produce the end result –this is the essence of functions. Now, to get to the idea of a set, one does not need stray from the conception of a group (in the conventional sense of the word). We group things together all the time. We group people, places, and things by attaching place-holders or by “tagging” a name to a group and then “baptizing” (see Kripke’s works, specifically, Naming and Necessity) the elements as members of the set we have tagged. An action such as ‘baptizing” an element a member of a specific group can be as arbitrary as one wants. For example, consider the group:

M={car, ~car, Fabio, train, shoe, glass, uranium, qualia, Hume, Leibniz, farm, Obama, Ross, jerk, nice and quarks}

Some of the objects in M share similarities, and some do not. The relations amid the elements in M are not consistent across all elements of the group M. Their are things in M that are inconsistent (i.e. car, and ~car or Hume and Leibniz). The properties shared among each elements, and across all elements are not uniform. Nonetheless, this is a set. That is, we say M is a set, and the stuff in M are its elements.

Cantor proclaimed,

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set

Some sets are better than others. I mean, consider the set of real numbers:

${R}$ ={0,1/2,2,6,600, 21, $\pi$ , e, 4, -5,…}

This set is very specific as to what elements can be contained in it. In fact, there exists a criteria, which is a list of properties that each element must possess in order to be in this set.

Sets are also acted upon by certain rules and operations. For example, what if we take a simple version of M, say M’ and let it possess the following:

M’={car, train, shoe}

and, now, let a new set called R contain the following:

R={car, leaf, stalk}

We can, from a linguistic standpoint, say, “hey, M’ and R, together, are the set of {car, train, shoe, leaf, stalk}!” This is true, and we can say that “the set M’ and R share {car}!” This is where the simple operation of disjunction and conjunction apply (i.e. or/and), but from a set-theroetical view, they do not necessarily apply in the exact same way we use the words “or” and “and.” In fact, as you can see above, putting the sets together is adding all of the elements of one to the other. This process calls for a disjunction or uniting of the sets as whole. Moreover, the process the entails a conjunction is one where one set intersects another set. So, we say,

M’ $\cup$ R ={car, leaf, stalk, train, shoe} for M’ united with R, and for M’ intersects R, we say that M’ $\cap$ R ={car}.

So, what, then, does it mean to say that there exists some $\phi$ , a set function on A?

2) Defintion: A set function is a function (same as above) whose input is a set and the output is a real number.

Basically, probability is a set function as is a measure (as we shall soon see).

If any ${A}$ and ${B}$ are two sets, we write ${A}-{B}$ to indicate that if ${x}\in{A}\rightarrow {x}\not\in{B}$ or ${A}\cup{B^c}$ where ${B^c}$ denotes the compliment of B. And if ${A}$ and ${B}$ are disjoint, we say that their intersection is empty -that is, they do not share anything. We denote such sets as ${A}\cap{B}=0$ .

Now, since we are trying to work our way down to a very fundamental and important idea in mathematics, we have to continue with some more definitions. Up to now, all this has seemed very basic and, perhaps, peace-wise. Thus, let’s pin-point our goal.

We are trying to develop the theory of the Lebesgue integral; this will require the expounding of set functions, rings, and, as such, the development of measure theory. From there, we arrive at Lebesgue integration.

Measure Theory

Definitions and properties

1) Defintion: A family ${\Re}$ of sets is called a ring if ${A}\in\Re$ and ${B}\in\Re$ $latex{\rightarrow}$

i) ${A}\cup{B}\in\Re$ and ${A}-{B}\in\Re$

ii) if $\Re$ is a ring, then ${A}\cap{B}$ is a ring.

A ring is called a ${\sigma}$ -ring if,

i) $\bigcup\limits_{n=1}^{\infty}A_n \in\Re$

whenever $A_n \in\Re (n=1,2,3,....).$ Since

ii) $\bigcap\limits_{n=1}^{\infty}A_n=A$

if ${\Re}$ is a ${\sigma}$ -ring.

This is all very nice, but it is very convoluted. Lets reduce it down a bit, shall we?

Let $X$ be any non-empty set. A $latex{\sigma}$-algebra of subsets of $X$ is a family ${\Re}$ of subsets of ${X}$ with the following properties:

p1) ${\Re}$ is non-empty.

p2) Closure under compliment: if ${B}\in\Re\rightarrow{A}\cup{B^c}\in\Re$ or, you could say, A\B is in ${\Re}$ .

p3) Closure under countable union: if ${B_n}$ is a sequence of sets in ${\Re}$ then $\bigcup\limits_{n=1}^{\infty}B_n\in\Re$

2) Definition: We say that ${\phi}$ is a set function defined on ${\Re}$ if it assigns to every $A\in\Re$ a number $latex\phi({A})$ of the real numbers line. Now, we say that ${\phi}$ is additive if ${A}\cap{B}=0$ implies:

p4) ${\phi}({A}\cap{B})=\phi{(A)}+\phi{(B)}$

Now, we really want to jump a bit further, and define ${\phi}$ as countable additive if ${A_i}\cap{A_j} =0$ for j not equal to i implies

p5) Countable additivity: for any mutually disjoint (see above) sets $A_n\in\Re$ ,

$\phi(\bigcup\limits_{n=1}^{\infty}A_n)=\sum\limits_{n=1}^{\infty}\phi(A_n)$

3) Theorem: Suppose that our set function $latex{\phi}$ is countably additive on a ring ${\Re}$ . Suppose further that $A_n\in\Re$ where n is a natural, $A_1\subset{A_2}\subset\cdots\subset{A_n}$ , ${A}\in\Re$ and,

$A=\bigcup\limits_{n=1}^{\infty}A_n$

then as $n\rightarrow\infty$ , we have:

$\phi{(A_n)}\rightarrow\phi{(A)}$

Proof:

Set $B_1 =A_1$ and notice that $B_n = A_n - A_{n-1}$ . It follows that, ${B_i}\cap{B_j}=0$ for ${i}\neq{j}$ , ${A_n} ={B_1} \cup {B_2}\cup \cdots \cup{B_n}$ and $A=\bigcup{B_n}$ . Thus,