Introduction

At this point, we have spent many lessons developing the Lebesgue integral.

A natural question should now be bothering you:

Why did mathematicians invent the Lebesgue integral at all?

After all, calculus already had integration.

For centuries mathematicians used the Riemann integral successfully.

It helped create:

• Classical mechanics
• Electromagnetism
• Differential equations
• Engineering

So why wasn’t it enough?

The answer is one of the most important stories in modern mathematics.

The Lebesgue integral was invented because the Riemann integral is fundamentally limited.

To understand why, we must compare the two approaches carefully.

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The Riemann Philosophy

Suppose:

$$f:[a,b]\to\mathbb R$$

The Riemann integral partitions the domain:

$$[a,b]$$

into small intervals:

$$[x_{i-1},x_i]$$

and approximates the area using rectangles.

The integral is defined by:

$$\int_a^b f(x),dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(\xi_i)\Delta x_i$$

where:

$$\xi_i\in[x_{i-1},x_i]$$

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What Riemann Measures

The Riemann integral groups points according to:

Where they are located.

We slice the x-axis.

Then we ask:

How high is the function above each slice?

This is a geometric viewpoint.

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Example

Consider:

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

on:

$$[0,1]$$

Partition:

$$[0,1]$$

into many tiny intervals.

Construct rectangles.

Take limits.

We obtain:

$$\int_0^1x^2,dx=\frac13$$

Everything works beautifully.

---

The Lebesgue Philosophy

Lebesgue reversed the process.

Instead of grouping points by position, he grouped points by function value.

Rather than asking:

Where is x?

he asked:

Where does f(x) lie?

This seemingly simple change transformed analysis.

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Visual Comparison

Riemann

Cut vertically.| | | | |

Partition the domain.

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Lebesgue

Cut horizontally.====== ====== ====== ======

Partition the range.

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This difference is the heart of the subject.

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A Function That Causes Trouble

Consider:

$$f(x)=\mathbf1_{\mathbb Q}(x)$$

on:

$$[0,1]$$

This is called the Dirichlet function.

It equals:

$$1$$

for rational numbers.

It equals:

$$0$$

for irrational numbers.

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Why Riemann Fails

Every interval contains:

• rational numbers
• irrational numbers

Therefore every interval contains values:

$$0$$

and:

$$1$$

The lower sum becomes:

$$0$$

The upper sum becomes:

$$1$$

No matter how fine the partition.

Thus:

$$f$$

is not Riemann integrable.

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Why Lebesgue Succeeds

The rational numbers satisfy:

$$\lambda(\mathbb Q)=0$$

Thus:

$$\int_0^1\mathbf1_{\mathbb Q}(x),d\lambda=0$$

The function is perfectly Lebesgue integrable.

This example shocked mathematicians when Lebesgue introduced his theory.

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What Really Matters?

The Riemann integral cares about:

• discontinuities

The Lebesgue integral cares about:

• measure

This is a profound difference.

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Riemann’s Criterion

A bounded function is Riemann integrable if and only if:

its set of discontinuities has measure zero.

This theorem is beautiful because it already hints at Lebesgue measure.

Even Riemann integration secretly depends on measure theory.

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Example

Consider:

$$f(x)=\begin{cases}
1,&x=\frac12\
0,&x\neq\frac12
\end{cases}$$

The function is discontinuous at one point.

Since:

$$\lambda({\tfrac12})=0$$

it is Riemann integrable.

Its integral equals:

$$0$$

Lebesgue agrees.

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Where Riemann Breaks Down

As analysis advanced, mathematicians encountered:

• Fourier series
• Probability theory
• Functional analysis
• PDEs

Functions became increasingly irregular.

The Riemann integral could not handle them.

Lebesgue integration could.

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Convergence Problems

Recall the sequence:

$$f_n(x)=n\mathbf1_{(0,\frac1n)}(x)$$

Each function satisfies:

$$\int_0^1f_n(x),dx=1$$

Pointwise:

$$f_n(x)\to0$$

This example exposed weaknesses in classical integration.

The convergence theorems of Lebesgue theory explain exactly what happens.

Riemann integration has no comparable framework.

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Why Probability Needed Lebesgue

Modern probability studies random variables.

Random variables are measurable functions.

Expectations are integrals:

$$E[X]=\int X,dP$$

Without Lebesgue integration:

• modern probability
• stochastic processes
• Bayesian statistics

would not exist in their current form.

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A More Powerful Class of Functions

Every Riemann integrable function is Lebesgue integrable.

However:

There exist Lebesgue integrable functions that are not Riemann integrable.

Symbolically:

$${\text{Riemann Integrable}}\subsetneq{\text{Lebesgue Integrable}}$$

The inclusion is strict.

Lebesgue theory truly extends Riemann theory.

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Why Measure Theory Came First

Many students wonder:

Why did we spend so much time studying sigma-algebras and measures?

Now the reason becomes clear.

The Lebesgue integral depends entirely on measure.

Without measure:

there is no Lebesgue integration.

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A Statistical Perspective

Suppose a probability density is:

$$p(x)$$

Then:

$$P(A)=\int_A p(x),dx$$

This is fundamentally a Lebesgue integral.

Every probability density function is secretly measure theory in disguise.

This is one reason statisticians eventually become measure theorists.

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Functional Analysis Connection

The spaces:

$$L^1$$

$$L^2$$

$$L^p$$

are all defined using the Lebesgue integral.

Without Lebesgue integration:

there would be no modern Hilbert space theory.

No Fourier analysis.

No quantum mechanics.

No operator algebras.

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Why Lebesgue Won

By the 1930s the mathematical community had largely adopted Lebesgue integration because it:

Handles More Functions

Including many pathological examples.

Has Better Convergence Theorems

MCT, Fatou, DCT.

Works Naturally with Probability

Expectation becomes integration.

Supports Functional Analysis

The language of modern analysis.

Generalizes Easily

To manifolds, operator algebras, and beyond.

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The Historical Impact

The creation of Lebesgue integration is often regarded as one of the most important events in twentieth-century mathematics.

It fundamentally changed:

• Analysis
• Probability
• Statistics
• Mathematical Physics

Almost everything we study later traces back to this development.

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Connection to Alain Connes

Connes often describes noncommutative geometry as:

A new form of measure theory and integration.

The progression is:

Classical Geometry

$$\longrightarrow$$

Measure Theory

$$\longrightarrow$$

Lebesgue Integration

$$\longrightarrow$$

Functional Analysis

$$\longrightarrow$$

Operator Algebras

$$\longrightarrow$$

Noncommutative Geometry

In many ways, Lebesgue’s revolution was the first step toward Connes’ revolution.

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Key Concepts Learned

By the end of this lesson you should understand:

• Riemann integration partitions the domain.
• Lebesgue integration partitions the range.
• Lebesgue integration handles more functions.
• The Dirichlet function is not Riemann integrable but is Lebesgue integrable.
• Every Riemann integrable function is Lebesgue integrable.
• Convergence theorems make Lebesgue integration far more powerful.
• Modern probability and functional analysis rely on Lebesgue integration.
• Measure theory was created largely to support this new notion of integration.

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Looking Ahead

In the next lesson:

Measure Theory Lesson 25: Convergence in Measure

we introduce a new notion of convergence that sits between pointwise convergence and convergence in probability. This concept becomes fundamental in probability theory, statistics, asymptotic analysis, and the study of $$L^p$$ spaces.