Introduction

In Lesson 9, we constructed the Lebesgue integral. We began with simple functions, approximated measurable functions from below, and used these approximations to define integration for a large class of functions.

A definition alone is not enough to make a mathematical theory useful. The true power of the Lebesgue integral comes from the remarkable properties it satisfies.

These properties allow us to:

• Manipulate integrals algebraically
• Compare functions through their integrals
• Establish bounds
• Develop probability theory
• Build functional analysis
• Eventually study operator algebras and noncommutative geometry

This lesson develops the fundamental properties of the Lebesgue integral.

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Property 1: Linearity

The most important property of integration is linearity.

Suppose that:

$$f,g \in L^1(\mu)$$

and:

$$a,b \in \mathbb{R}$$

Then:

$$\int (af+bg),d\mu = a\int f,d\mu + b\int g,d\mu$$

This means that integration distributes over addition and scalar multiplication.

Example

Suppose:

$$\int f,d\mu = 5$$

and:

$$\int g,d\mu = 3$$

Then:

$$\int (2f-4g),d\mu = 2(5)-4(3)=-2$$

Notice that we never needed to know the functions themselves. Knowing their integrals was sufficient.

Why It Matters

Linearity is the foundation of:

• Fourier analysis
• Functional analysis
• Quantum mechanics
• Differential equations
• Probability theory

Without linearity, modern analysis would be impossible.

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Property 2: Positivity

Suppose:

$$f(x)\ge0$$

for every:

$$x\in X$$

Then:

$$\int f,d\mu \ge 0$$

This property aligns perfectly with intuition.

If a function never becomes negative, its total accumulated mass cannot be negative.

Example

Since:

$$x^2\ge0$$

for every real number:

$$x$$

we must have:

$$\int_{-1}^{1}x^2,dx \ge0$$

Indeed:

$$\int_{-1}^{1}x^2,dx=\frac{2}{3}$$

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Property 3: Monotonicity

Suppose:

$$f(x)\le g(x)$$

for every:

$$x\in X$$

Then:

$$\int f,d\mu \le \int g,d\mu$$

Larger functions produce larger integrals.

Example

On the interval:

$$[0,1]$$

we have:

$$x^2\le x$$

Therefore:

$$\int_0^1x^2,dx \le \int_0^1x,dx$$

Indeed:

$$\frac13 \le \frac12$$

Monotonicity allows us to compare integrals without explicitly computing them.

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Property 4: Absolute Value Inequality

One of the most useful inequalities in analysis is:

$$\left|\int f,d\mu\right| \le \int |f|,d\mu$$

This says:

The magnitude of the average cannot exceed the average magnitude.

Example

Suppose:

$$\int |f|,d\mu = 20$$

Then automatically:

$$\left|\int f,d\mu\right|\le20$$

Even without knowing the function, we immediately obtain a bound.

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Property 5: Integrating Over Sets

Often we wish to integrate over only part of a space.

For a measurable set:

$$A$$

we define:

$$\int_A f,d\mu = \int f\mathbf{1}_A,d\mu$$

where:

$$\mathbf{1}_A$$

is the indicator function of:

$$A$$

This simple idea allows integration over arbitrary measurable regions.

Example

If:

$$f(x)=x$$

then:

$$\int_{[0,1]}x,dx=\frac12$$

while:

$$\int_{[0,2]}x,dx=2$$

The region being integrated over matters.

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Property 6: Additivity Over Disjoint Sets

Suppose:

$$A\cap B=\emptyset$$

Then:

$$\int_{A\cup B}f,d\mu = \int_A f,d\mu + \int_B f,d\mu$$

This mirrors countable additivity of measures.

Example

Suppose:

$$f(x)=1$$

on:

$$[0,2]$$

Then:

$$\int_{[0,2]}1,dx=\int_{[0,1]}1,dx+\int_{[1,2]}1,dx$$

which becomes:

$$2=1+1$$

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Property 7: Measure Zero Sets Do Not Matter

This is one of the deepest ideas in measure theory.

Suppose:

$$f=g$$

everywhere except on a set of measure zero.

Then:

$$\int f,d\mu=\int g,d\mu$$

The integral cannot detect differences occurring only on null sets.

Example

Define:

$$f(x)=0$$

for every:

$$x$$

and:

$$g(x)=\begin{cases}
1,&x=0\
0,&x\neq0
\end{cases}$$

The functions differ at exactly one point.

Since:

$$m({0})=0$$

we obtain:

$$\int f,dm=\int g,dm=0$$

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Almost Everywhere

Because measure-zero sets are invisible to integration, we introduce the notion of almost everywhere.

We write:

$$f=g \quad \text{a.e.}$$

if:

$$\mu({x:f(x)\neq g(x)})=0$$

In measure theory, functions equal almost everywhere are often treated as identical.

This idea becomes crucial when studying:

• $$L^p$$ spaces
• Probability theory
• Functional analysis
• Operator algebras

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Property 8: Bounded Functions on Finite Measure Sets

Suppose:

$$|f(x)|\le M$$

for all:

$$x\in A$$

and:

$$\mu(A)<\infty$$

Then:

$$\int_A |f|,d\mu \le M\mu(A)$$

This provides an immediate upper bound.

Example

If:

$$|f(x)|\le5$$

and:

$$\mu(A)=4$$

then:

$$\int_A|f|,d\mu\le20$$

Such estimates appear constantly in analysis.

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The Emergence of Functional Analysis

Notice how the integral is beginning to behave like an algebraic object.

To every function:

$$f$$

we assign a number:

$$\int f,d\mu$$

and this assignment is linear.

In modern language, integration is a linear functional.

This observation is one of the first bridges from measure theory to functional analysis.

Many years later, Connes will replace functions by operators and study generalized linear functionals called traces.

The seeds of that theory are already visible here.

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Why These Properties Matter

At first glance, these properties seem technical.

In reality, they are what make the Lebesgue integral useful.

They allow us to:

• Manipulate integrals symbolically
• Establish inequalities
• Build probability theory
• Define expectations
• Construct Hilbert spaces
• Develop operator theory

Without these properties, the Lebesgue integral would merely be a definition.

With them, it becomes one of the most powerful tools in mathematics.

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The Big Question Ahead

The next major challenge is understanding limits.

Suppose:

$$f_n \to f$$

as:

$$n\to\infty$$

Can we conclude:

$$\lim_{n\to\infty}\int f_n,d\mu = \int f,d\mu$$

Sometimes yes.

Sometimes no.

The answer depends on how convergence occurs.

Understanding this question leads directly to:

• Modes of convergence
• Monotone Convergence Theorem
• Fatou’s Lemma
• Dominated Convergence Theorem

These are among the most important results in all of analysis.

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Key Concepts Learned

By the end of this lesson you should understand:

• The Lebesgue integral is linear.
• Nonnegative functions have nonnegative integrals.
• Larger functions have larger integrals.
• The inequality $$\left|\int f,d\mu\right| \le \int |f|,d\mu$$ provides a powerful bound.
• Integrals are additive over disjoint sets.
• Functions equal almost everywhere have the same integral.
• Measure-zero sets are invisible to integration.
• Integration is a linear functional, foreshadowing functional analysis and operator theory.

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Looking Ahead

In Lesson 11 we begin one of the deepest parts of measure theory:

Modes of Convergence.

A sequence of functions can converge in many different ways, and understanding these distinctions is essential before proving the great convergence theorems that make Lebesgue integration so powerful.