From Measuring Sets to Measuring Functions

So far, measure theory has focused on sets:

• measurable sets
• sigma-algebras
• measures
• Lebesgue measure

However, the ultimate goal of measure theory is not merely to measure sets.

The ultimate goal is to integrate functions.

Before we can define integration, we must answer a fundamental question:

Which functions are compatible with the measurable structure of a space?

The answer leads to one of the most important concepts in all of analysis:

Measurable functions.

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Why We Need Measurable Functions

Suppose:

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

assigns a real number to each point of a measurable space:

$$\left(X,\mathcal{F}\right)$$

For example:

$$f(x)=x^2$$

or

$$f(x)=\sin(x)$$

or

$$f(x)=\mathbf{1}_A(x)$$

where:

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

To integrate a function, we must ensure it behaves well with respect to measurable sets.

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Intuition

A measurable function is one that does not destroy measurability.

If we start with measurable subsets of the output space,

their preimages should also be measurable.

In other words:

Measurable sets should pull back to measurable sets.

This is the measure-theoretic analogue of continuity in topology.

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Formal Definition

Let:

$$\left(X,\mathcal{F}\right)$$

and

$$\left(Y,\mathcal{G}\right)$$

be measurable spaces.

A function:

$$f:X\to Y$$

is measurable if:

$$f^{-1}(B)\in\mathcal{F}$$

for every:

$$B\in\mathcal{G}$$

Here:

$$f^{-1}(B)={x\in X:f(x)\in B}$$

is called the preimage of:

$$B$$

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Why Preimages?

Notice something important.

The definition says nothing about:

$$f(B)$$

It only talks about:

$$f^{-1}(B)$$

Preimages behave much better mathematically than images.

For example:

Preimages automatically preserve:

• unions
• intersections
• complements

This makes them ideal for measure theory.

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Example 1

Consider:

$$f(x)=x^2$$

on:

$$\mathbb{R}$$

Take:

$$B=(1,4)$$

Then:

$$f^{-1}(B)=(-2,-1)\cup(1,2)$$

This set is measurable.

In fact every open interval has a measurable preimage.

Therefore:

$$f(x)=x^2$$

is measurable.

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Example 2

Consider:

$$f(x)=\sin(x)$$

Take:

$$B=(0,1)$$

The preimage consists of infinitely many intervals:

$$\bigcup_{k\in\mathbb{Z}}
(2k\pi,(2k+1)\pi)
$$

This is measurable.

Therefore:

$$\sin(x)$$

is measurable.

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Example 3: Constant Functions

Suppose:

$$f(x)=c$$

for every:

$$x$$

Take any measurable set:

$$B$$

If:

$$c\in B$$

then:

$$f^{-1}(B)=X$$

If:

$$c\notin B$$

then:

$$f^{-1}(B)=\emptyset$$

Both sets are measurable.

Therefore every constant function is measurable.

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Indicator Functions

One of the most important measurable functions is the indicator function.

For a measurable set:

$$A$$

define:

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

This function encodes the set itself.

In many ways:

Sets and indicator functions are interchangeable.

This observation becomes crucial when building the Lebesgue integral.

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Measurable Functions on the Real Line

When:

$$Y=\mathbb{R}$$

with the Borel sigma-algebra,

the definition simplifies dramatically.

A function:

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

is measurable if:

$${x:f(x)>a}$$

is measurable for every real number:

$$a$$

This criterion is used constantly in practice.

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Equivalent Conditions

The following are all equivalent.

A real-valued function is measurable if:

$${x:f(x)>a}$$

is measurable for every:

$$a$$

or

$${x:f(x)\ge a}$$

is measurable for every:

$$a$$

or

$${x:f(x)<a}$$

is measurable for every:

$$a$$

or

$${x:f(x)\le a}$$

is measurable for every:

$$a$$

Any one of these is sufficient.

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Continuous Functions Are Measurable

A crucial theorem states:

Every continuous function is measurable.

Examples:

$$x^2$$

$$\sin(x)$$

$$e^x$$

$$\log(x)$$

are all measurable.

Thus measurable functions include all familiar functions from calculus.

However, the class of measurable functions is much larger than the class of continuous functions.

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Closure Properties

Measurable functions behave very nicely.

Suppose:

$$f$$

and

$$g$$

are measurable.

Then:

Sum

$$f+g$$

is measurable.

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Difference

$$f-g$$

is measurable.

---

Product

$$fg$$

is measurable.

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Quotient

$$\frac{f}{g}$$

is measurable wherever:

$$g\neq0$$

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Maximum

$$\max(f,g)$$

is measurable.

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Minimum

$$\min(f,g)$$

is measurable.

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These properties allow us to construct increasingly complicated measurable functions.

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Why Measurable Functions Matter

Measurable functions are to measure theory what continuous functions are to topology.

Without them:

• integration cannot be defined
• probability theory cannot exist
• stochastic processes cannot be studied

They are the basic objects upon which modern analysis operates.

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Random Variables Are Measurable Functions

This is one of the most important facts in probability.

A random variable is simply:

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

that is measurable.

Nothing more.

Thus:

Random variables are measurable functions.

Modern probability theory is measure theory applied to measurable functions.

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A Geometric Interpretation

Think of a measurable function as respecting the observable structure of a space.

If a set of outputs can be observed,

then the corresponding inputs producing those outputs must also be observable.

This ensures compatibility between:

• the geometry of the space
• the measure defined on it

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Connection to Alain Connes

In classical measure theory:

• measurable sets are observable events
• measurable functions are observable quantities

In noncommutative geometry, observables become operators.

One can think of measurable functions as the first step toward operator algebras.

The progression is roughly:

$$\text{Sets}
\rightarrow
\text{Functions}
\rightarrow
\text{Operators}
$$

Much of Connes’ work begins by replacing ordinary functions with operator-theoretic objects.

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Key Concepts Learned

By the end of this lesson you should understand:

• A measurable function preserves measurability through preimages.
• The formal definition involves measurable preimages.
• Constant functions are measurable.
• Indicator functions are measurable.
• Continuous functions are measurable.
• Sums, products, and limits of measurable functions remain measurable.
• Random variables are measurable functions.
• Measurable functions are the fundamental objects integrated by the Lebesgue integral.

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Looking Ahead

In the next lesson:

Lesson 8: Simple Functions

we will construct the building blocks of the Lebesgue integral. Just as polygons approximate curves in geometry, simple functions approximate measurable functions and allow integration to be built from first principles. This is the key idea behind Lebesgue’s revolutionary approach to integration.