Why We Need Simple Functions

In calculus, integration begins with rectangles.

To find the area under a curve, we approximate the curve using many small rectangles.

Lebesgue had a similar idea, but instead of approximating geometry, he approximated functions.

His question was:

What is the simplest type of measurable function we can integrate exactly?

The answer is the simple function.

Simple functions are the atoms from which the entire Lebesgue integral is built.

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Definition of a Simple Function

A measurable function:

$$\phi : X \to \mathbb{R}$$

is called a simple function if it takes only finitely many values.

In other words:

$$\phi(X)=\{a_1,a_2,\ldots,a_n\}$$

for some finite collection of numbers.

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Example 1

Consider:

$$\phi(x)=\begin{cases}
1,&x<0\\
2,&x\ge0
\end{cases}$$

This function only takes two values:

$$1 \text{ and } 2$$

Therefore it is a simple function.

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Example 2

Consider:

$$\phi(x)=\begin{cases}
0,&x<0\\
1,&0\le x<1\\
3,&x\ge1
\end{cases}$$

The values are:

$${0,1,3}$$

Only three values occur.

Thus it is simple.

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Example 3

The indicator function:

$$\mathbf{1}_A(x)=\begin{cases}
1,&x\in A\\
0,&x\notin A
\end{cases}$$

takes only:

$$0$$

and

$$1$$

Therefore every indicator function is a simple function.

Indicator functions are the most fundamental simple functions.

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Standard Representation

Every nonnegative simple function can be written as:

$$\phi(x)=\sum_{i=1}^{n} a_i \mathbf{1}_{A_i}(x)$$

where:

$$a_i\ge0$$

each $$A_i$$ is measurable and the sets are pairwise disjoint¹

This representation is extremely important.

It says:

Every simple function is a weighted combination of indicator functions.

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Example

Suppose:

$$\phi(x)=\begin{cases}
2,&x\in A\\
5,&x\in B\\
0,&\text{otherwise}
\end{cases}$$

where:

$$A\cap B=\emptyset$$

Then:

$$\phi=2\mathbf{1}_A+5\mathbf{1}_B$$

This is the standard representation.

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Why Simple Functions Matter

Notice something remarkable.

If we know:

• the values $$a_i$$
• the sizes of the sets $$A_i$$

then we already know how much “mass” the function contains.

This suggests a natural definition of integration.

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Integrating an Indicator Function

Suppose A is measurable.

The indicator function is:

$$\mathbf{1}_A$$

Since the function equals: 1 on A and 0 elsewhere, its integral should simply equal the size of A.

Therefore:

$$\int \mathbf{1}_A,d\mu=\mu(A)$$

This is the foundation of everything that follows.

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Integrating a Simple Function

Suppose:

$$\phi=\sum_{i=1}^{n}a_i\mathbf{1}_{A_i}$$

Then we define:

This definition is incredibly natural.

Each value:

$$a_i$$

is weighted by the measure of the region where it occurs.

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Example 1

Suppose:

$$\phi=3\mathbf{1}_A$$

and:

$$\mu(A)=4$$

Then:

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Example 2

Suppose:

with:

$$\mu(A)=3$$

and

$$\mu(B)=2$$

Then:

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A Geometric Interpretation

For a simple function:

the integral becomes:

$$\sum_{i=1}^{n}
a_i\mu(A_i)
$$

This is exactly the volume of a collection of rectangular blocks.

Each block has:

• height $$a_i$$
• base measure $$\mu(A_i)$$

Thus simple-function integration generalizes the rectangle rule from calculus.

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Approximating Complicated Functions

Now comes Lebesgue’s key insight.

Consider the function:

$$f(x)=x^2,\quad x\in[0,1]$$

This is not a simple function because it takes infinitely many values. Recall that a simple function can only take finitely many values.

Lebesgue’s idea was to approximate complicated measurable functions by simple functions.

For example, we can define the simple function:

This is a very crude approximation of$\ f(x)=X^2$

A better approximation is:

This simple function takes only three values:

We can improve the approximation further by dividing the interval into more pieces and assigning a constant value on each piece.

For example, we might partition ($[0,1]$) into ten intervals and define a step function that approximates ($x^2$) on each interval.

As the number of steps increases, the approximation becomes more accurate. The graph begins to resemble the smooth curve ($f(x)=x^2$) more and more closely.

The key idea is that every nonnegative measurable function can be approximated from below by an increasing sequence of simple functions:

Lebesgue then defines the integral of (f) by looking at the integrals of these simple approximations and taking the supremum:

This is one of the fundamental ideas of measure theory. Instead of integrating a complicated function directly, we approximate it using simple functions whose integrals are easy to compute. The Lebesgue integral is then obtained as the limit of these increasingly accurate approximations.

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Building the Lebesgue Integral

Lebesgue’s revolutionary idea was:

1. Integrate simple functions exactly.
2. Approximate measurable functions by simple functions.
3. Define the integral of the measurable function as the limit.

Thus the entire Lebesgue integral rests on simple functions.

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Approximation from Below

For a nonnegative measurable function:

$$f$$

we construct simple functions:

$$\phi_1\le\phi_2\le\phi_3\le\cdots\le f$$

such that:

$$\phi_n(x)\to f(x)$$

for every:

$$x$$

The sequence rises toward:

$$f$$

from below.

This idea is central to Lebesgue integration.

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Why This Is Better Than Riemann’s Approach

Riemann approximates the domain.

Lebesgue approximates the function values.

Riemann:

• divide the x-axis

Lebesgue:

• group together points having similar function values

This subtle change makes many previously impossible integrals manageable.

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Simple Functions in Probability

Suppose:

$$X$$

is a random variable.

A simple random variable might be:

$$X=\begin{cases}
100,&\text{with probability }0.3\\
200,&\text{with probability }0.7
\end{cases}$$

Then:

$$E[X]=100(0.3)+200(0.7)=170$$

Notice how expectation is exactly the same formula as the integral of a simple function. This is not a coincidence. Expectation is a Lebesgue integral.

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Connection to Alain Connes

Simple functions are the first step in a profound progression:

$$\text{Indicator Functions}
\rightarrow
\text{Simple Functions}
\rightarrow
\text{Measurable Functions}
\rightarrow
\text{Operators}
$$

Classical measure theory builds integration from indicator functions.

Connes eventually replaces functions with operators and develops generalized notions of integration on noncommutative spaces.

Understanding simple functions is therefore understanding the first brick in a much larger structure.

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Key Concepts Learned

By the end of this lesson you should understand:

• A simple function takes only finitely many values.
• Every simple function can be written using indicator functions.
• Indicator functions are the building blocks of simple functions.
• The integral of a simple function is:

• Measurable functions can be approximated by simple functions.
• Lebesgue integration is built from these approximations.
• Expectation in probability is a special case of Lebesgue integration.

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Looking Ahead

In the next lesson:

Lesson 9: Constructing the Lebesgue Integral

we will take the final step from simple functions to arbitrary nonnegative measurable functions and formally define the Lebesgue integral, one of the most important constructions in all of modern mathematics.

1. Pairwise disjoint means that every two distinct sets in a collection have no elements in common. In other words, if you pick any two different sets from the collection, their intersection is empty. For example, the sets $A=\{1,2\}, B=\{3,4\}$, and $C=\{5,6\}$ are pairwise disjoint because $A\cap B=\emptyset, A\cap C=\emptyset, and\  B\cap C=\emptyset$. This concept is impozrtant in measure theory because measures are additive on pairwise disjoint sets: the measure of their union equals the sum of their individual measures. ↩︎