Introduction

In the previous lesson, we proved the Monotone Convergence Theorem.

The theorem stated that if:

$$f_n\uparrow f$$

and all functions are nonnegative, then:

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

This was a powerful result, but it required a strong assumption:

the sequence had to be increasing.

A natural question arises:

What happens when the sequence is not monotone?

Can we still relate limits and integrals?

The answer is yes, although not with an equality.

Instead, we obtain one of the most important inequalities in analysis:

Fatou’s Lemma.

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The Problem

Suppose:

$$f_1,f_2,f_3,\ldots$$

are nonnegative measurable functions.

We wish to compare:

$$\int \liminf_{n\to\infty} f_n,d\mu$$

with:

$$\liminf_{n\to\infty}\int f_n,d\mu$$

At first glance these expressions may look intimidating.

The key idea is understanding the quantity:

$$\liminf_{n\to\infty}f_n$$

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What Is the Limit Inferior?

For a sequence of numbers:

$$a_1,a_2,a_3,\ldots$$

the limit inferior is the eventual lower limit of the sequence.

It is denoted:

$$\liminf_{n\to\infty}a_n$$

Intuitively:

Ignore finitely many early terms and look at the smallest values that keep appearing infinitely often.

For functions, we apply this pointwise.

Thus:

$$\liminf_{n\to\infty}f_n(x)$$

is computed separately for each:

$$x$$

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Example

Consider:

$$1,0,1,0,1,0,\ldots$$

The sequence never settles.

However:

$$\liminf_{n\to\infty}=0$$

because arbitrarily far along the sequence we continue to encounter zeros.

Similarly:

$$\limsup_{n\to\infty}=1$$

because arbitrarily far along the sequence we continue to encounter ones.

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Statement of Fatou’s Lemma

Let:

$$f_n:X\to[0,\infty]$$

be nonnegative measurable functions.

Then:

$$\int \liminf_{n\to\infty}f_n,d\mu\le\liminf_{n\to\infty}\int f_n,d\mu$$

This is Fatou’s Lemma.

It is one of the fundamental results of measure theory.

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Why the Inequality Points This Way

Many students initially guess the opposite inequality.

However, Fatou’s Lemma says:

$$\text{Integral of the limit inferior} \le \text{Limit inferior of the integrals}$$

The right-hand side can contain “extra mass” that disappears in the limit.

Therefore it is usually larger.

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Intuition

Imagine a sequence of landscapes.

Each:

$$f_n$$

represents the height of the landscape.

The limit inferior captures the part of the landscape that survives indefinitely.

Transient peaks eventually disappear.

The integral of the limit inferior measures only the surviving mass.

The integrals of:

$$f_n$$

still count temporary peaks.

Therefore the right side can be larger.

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A Key Construction

Define:

$$g_n(x)=\inf_{k\ge n}f_k(x)$$

For every:

$$x$$

the sequence:

$$g_n(x)$$

is increasing.

Moreover:

$$g_n(x)\uparrow \liminf_{n\to\infty}f_n(x)$$

This is the crucial observation behind the proof.

By converting an arbitrary sequence into an increasing one, we can use the Monotone Convergence Theorem.

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Sketch of the Proof

Define:

$$g_n(x)=\inf_{k\ge n}f_k(x)$$

Then:

$$g_n\uparrow \liminf_{n\to\infty}f_n$$

Applying MCT:

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

Since:

$$g_n\le f_k$$

for every:

$$k\ge n$$

we obtain:

$$\int g_n,d\mu\le\inf_{k\ge n}\int f_k,d\mu$$

Taking limits gives:

$$\int \liminf_{n\to\infty}f_n,d\mu\le\liminf_{n\to\infty}\int f_n,d\mu$$

which proves the theorem.

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Example 1

Suppose:

$$f_n(x)=\mathbf{1}_{[\frac1n,1]}(x)$$

on:

$$[0,1]$$

As:

$$n\to\infty$$

the functions converge pointwise to:

$$f(x)=\mathbf{1}_{(0,1]}(x)$$

The integrals are:

$$\int_0^1f_n(x),dx=1-\frac1n$$

Therefore:

$$\liminf_{n\to\infty}\int_0^1f_n(x),dx=1$$

The limiting function has integral:

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

Hence:

$$1\le1$$

and equality holds.

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Example 2

Consider:

$$f_n(x)=n\mathbf{1}_{(0,\frac1n)}(x)$$

on:

$$[0,1]$$

We saw this example earlier.

Pointwise:

$$f_n(x)\to0$$

for every:

$$x>0$$

Thus:

$$\liminf f_n=0$$

and:

$$\int_0^1\liminf f_n,dx=0$$

However:

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

for every:

$$n$$

Therefore:

$$0\le1$$

The inequality is strict.

This example demonstrates why equality cannot be expected in general.

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Why Fatou’s Lemma Is Powerful

Fatou’s Lemma requires only:

1. Measurability
2. Nonnegativity

There is no requirement that:

• the sequence converges
• the sequence is monotone
• the sequence is bounded

This makes the theorem extraordinarily flexible.

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Lower Semicontinuity of the Integral

Fatou’s Lemma is often summarized as:

Integration is lower semicontinuous.

The integral of the limit inferior cannot exceed the limit inferior of the integrals.

This viewpoint becomes extremely important in optimization, variational analysis, and functional analysis.

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Fatou’s Lemma in Probability

Suppose:

$$X_n\ge0$$

are random variables.

Then:

$$E[\liminf X_n]\le\liminf E[X_n]$$

This result appears constantly in:

• stochastic processes
• Bayesian statistics
• asymptotic theory
• statistical learning

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Relationship to MCT

The Monotone Convergence Theorem gives:

$$\int \lim f_n,d\mu=\lim \int f_n,d\mu$$

under strong assumptions.

Fatou’s Lemma weakens the assumptions but also weakens the conclusion:

$$\int \liminf f_n,d\mu\le\liminf \int f_n,d\mu$$

Thus Fatou’s Lemma is more general.

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Why Analysts Love Fatou’s Lemma

Many difficult proofs proceed as follows:

1. Construct a sequence of approximations.
2. Apply Fatou’s Lemma.
3. Obtain an inequality.
4. Refine the argument further.

The theorem appears so frequently that experienced analysts often recognize situations where Fatou’s Lemma is hiding behind the scenes.

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Connection to Alain Connes

Fatou’s Lemma expresses a deep principle:

Integration behaves well under limits.

This principle reappears throughout functional analysis.

When Connes studies traces on operator algebras, analogous continuity and semicontinuity properties become essential.

Many sophisticated results in noncommutative integration can be viewed as extensions of ideas first encountered here.

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Key Concepts Learned

By the end of this lesson you should understand:

• The limit inferior captures the eventual lower behavior of a sequence.
• Fatou’s Lemma applies to nonnegative measurable functions.
• The theorem states:

$$\int \liminf_{n\to\infty}f_n,d\mu\le\liminf_{n\to\infty}\int f_n,d\mu$$

• Equality is not guaranteed.
• Fatou’s Lemma is more general than MCT.
• The theorem expresses lower semicontinuity of integration.
• It is one of the most frequently used tools in modern analysis.

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Looking Ahead

In the next lesson we reach one of the crown jewels of measure theory:

Lesson 14: The Dominated Convergence Theorem

The Dominated Convergence Theorem provides conditions under which limits and integrals can be exchanged for general sequences of functions. It is arguably the single most useful theorem in all of Lebesgue integration and appears throughout probability theory, statistics, functional analysis, and mathematical physics.