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RA-3

8-12 hours

Advanced

Lebesgue Integration Core & L1

Master the Lebesgue integral for simple, nonnegative, and general measurable functions. Understand absolute integrability, L1 space structure, and fundamental convergence theorems including Fatou's lemma and monotone convergence.

Learning Objectives

By the end of this course, you will be able to:

Define the Lebesgue integral for simple functions and extend to nonnegative measurable functions

Understand absolute integrability and define the integral for general measurable functions

Master Fatou's lemma and the monotone convergence theorem

Understand the structure of L1 space as a complete normed vector space

Compare Lebesgue integration with Riemann integration

Apply integration theory to solve problems involving limits and integrals

Prerequisites

Before starting this course, you should be familiar with:

Measurable functions and simple function approximations
Measure theory: measurable sets, outer measure, Lebesgue measure
Basic properties of sequences and limits
Riemann integration (for comparison)

Core Concepts

Fundamental definitions and properties

Definition 3.1: Lebesgue Integral of Simple Functions

Let $\varphi = \sum_{i=1}^n c_i \chi_{E_i}$ be a simple function, where $E_i$ are disjoint measurable sets. The Lebesgue integral of $\varphi$ is defined as:

\int \varphi \, dm = \sum_{i=1}^n c_i m(E_i)

This definition is independent of the representation of $\varphi$ .

Definition 3.2: Lebesgue Integral of Nonnegative Functions

For a nonnegative measurable function $f$ , the Lebesgue integral is defined as:

\int f \, dm = \sup\left\{\int \varphi \, dm : \varphi \text{ simple}, 0 \leq \varphi \leq f\right\}

The integral may be infinite. If $\int f \, dm < \infty$ , we say $f$ is integrable.

Definition 3.3: Absolute Integrability and L1

A measurable function $f$ is said to be absolutely integrable (or in $L^1$ ) if:

\int |f| \, dm < \infty

The space $L^1(\mathbb{R}^n)$ consists of all absolutely integrable functions (modulo functions that are zero almost everywhere).

Definition 3.4: Lebesgue Integral of General Functions

For a general measurable function $f$ , define the positive and negative parts:

f^+(x) = \max(f(x), 0), \quad f^-(x) = \max(-f(x), 0)

Then $f = f^+ - f^-$ and $|f| = f^+ + f^-$ . The Lebesgue integral is defined as:

\int f \, dm = \int f^+ \, dm - \int f^- \, dm

provided at least one of the integrals on the right is finite. If both are finite, $f \in L^1$ .

Theorem 3.1: Linearity of the Integral

Let $f, g$ be integrable functions and $\alpha, \beta \in \mathbb{R}$ . Then:

\int (\alpha f + \beta g) \, dm = \alpha \int f \, dm + \beta \int g \, dm

Proof of Theorem 3.1:

Proof:

The proof proceeds by first establishing linearity for simple functions, then extending to nonnegative functions using the definition, and finally to general functions using the positive/negative decomposition. ∎

∎

Theorem 3.2: Fatou's Lemma

Let $\{f_n\}$ be a sequence of nonnegative measurable functions. Then:

\int \liminf_{n \to \infty} f_n \, dm \leq \liminf_{n \to \infty} \int f_n \, dm

Proof of Theorem 3.2:

Proof:

Let $g_n = \inf_{k \geq n} f_k$ . Then $\{g_n\}$ is an increasing sequence with $g_n \to \liminf f_n$ .

By the monotone convergence theorem (which we prove next), $\int g_n \to \int \liminf f_n$ .

Since $g_n \leq f_n$ , we have $\int g_n \leq \int f_n$ , so

\int \liminf f_n = \lim_{n \to \infty} \int g_n \leq \liminf_{n \to \infty} \int f_n

∎

Theorem 3.3: Monotone Convergence Theorem

Let $\{f_n\}$ be an increasing sequence of nonnegative measurable functions converging pointwise to $f$ . Then:

\int f \, dm = \lim_{n \to \infty} \int f_n \, dm

Proof of Theorem 3.3:

Proof:

Since $f_n \leq f$ , we have $\int f_n \leq \int f$ , so $\lim \int f_n \leq \int f$ .

For the reverse inequality, let $\varphi$ be a simple function with $0 \leq \varphi \leq f$ . For $0 < \alpha < 1$ , define $E_n = \{x : f_n(x) \geq \alpha \varphi(x)\}$ .

Then $E_n \uparrow X$ and $\int f_n \geq \int_{E_n} f_n \geq \alpha \int_{E_n} \varphi$ .

Taking limits, $\lim \int f_n \geq \alpha \int \varphi$ . Since this holds for all $\alpha < 1$ and all simple $\varphi \leq f$ , we get $\lim \int f_n \geq \int f$ . ∎

∎

Example 3.1: Integral of a Characteristic Function

Problem: Compute $\int \chi_{[0,1]} \, dm$ on $\mathbb{R}$ .

Solution:

Since $\chi_{[0,1]}$ is a simple function,

\int \chi_{[0,1]} \, dm = 1 \cdot m([0,1]) = 1

This matches our intuitive notion that the integral should equal the length of the interval.

Example 3.2: Integral of a Step Function

Problem: Compute $\int f \, dm$ where $f(x) = 2\chi_{[0,1]}(x) + 3\chi_{(1,2]}(x)$ .

Solution:

\int f \, dm = 2 \cdot m([0,1]) + 3 \cdot m((1,2]) = 2 \cdot 1 + 3 \cdot 1 = 5

Corollary 3.1: L1 is a Vector Space

The space $L^1(\mathbb{R}^n)$ is a vector space under pointwise addition and scalar multiplication.

Theorem 3.4: Monotonicity and Comparison Properties of the Integral

Let $f$ and $g$ be measurable functions.

If $f \leq g$ a.e. and both integrals exist, then $\int f \, dm \leq \int g \, dm$
If $f \geq 0$ a.e. and $\int f \, dm = 0$ , then $f = 0$ a.e.
If $|f| \leq g$ a.e. and $g \in L^1$ , then $f \in L^1$
$\left|\int f \, dm\right| \leq \int |f| \, dm$ (when the integrals exist)

Proof of Theorem 3.4:

Proof:

(1) For nonnegative functions, this follows from the definition. For general functions, write $f = f^+ - f^-$ and $g = g^+ - g^-$ . Since $f \leq g$ , we have $f^+ \leq g^+$ and $f^- \geq g^-$ , so

\int f = \int f^+ - \int f^- \leq \int g^+ - \int g^- = \int g

(2) If $f \geq 0$ and $\int f = 0$ , then for any simple function $\varphi$ with $0 \leq \varphi \leq f$ , we have $\int \varphi = 0$ . This implies that $f$ is zero except on a null set.

(3) Since $|f| \leq g$ and $g \in L^1$ , we have $\int |f| \leq \int g < \infty$ , so $f \in L^1$ .

(4) We have $-|f| \leq f \leq |f|$ , so by (1), $-\int |f| \leq \int f \leq \int |f|$ , which gives the result. ∎

∎

Theorem 3.5: Riesz-Fischer Theorem: Completeness of L1

The space $L^1(\mathbb{R}^n)$ is complete with respect to the $L^1$ norm $\|f\|_1 = \int |f| \, dm$ . That is, every Cauchy sequence in $L^1$ converges to a function in $L^1$ .

Proof of Theorem 3.5:

Proof:

Let $\{f_n\}$ be a Cauchy sequence in $L^1$ . Then there exists a subsequence $\{f_{n_k}\}$ such that

\|f_{n_{k+1}} - f_{n_k}\|_1 < \frac{1}{2^k}

Define $g_k = |f_{n_1}| + \sum_{j=1}^{k-1} |f_{n_{j+1}} - f_{n_j}|$ . Then $\{g_k\}$ is an increasing sequence of nonnegative functions, and

\int g_k \leq \|f_{n_1}\|_1 + \sum_{j=1}^{k-1} \frac{1}{2^j} < \infty

By the monotone convergence theorem, $g_k \to g$ pointwise a.e. for some $g \in L^1$ .

Since $|f_{n_k}| \leq g_k \leq g$ , the sequence $\{f_{n_k}\}$ converges pointwise a.e. to some function $f$ . By Fatou's lemma,

\int |f - f_{n_k}| \leq \liminf_{j \to \infty} \int |f_{n_j} - f_{n_k}|

Since $\{f_n\}$ is Cauchy, the right-hand side tends to 0 as $k \\to \\infty$ , so $f_{n_k} \to f$ in $L^1$ . Since $\{f_n\}$ is Cauchy, the whole sequence converges to $f$ in $L^1$ . ∎

∎

Theorem 3.6: Absolute Continuity of the Integral

Let $f \in L^1$ . Then for every $\epsilon > 0$ , there exists $\delta > 0$ such that for any measurable set $E$ with $m(E) < \delta$ , we have

\int_E |f| \, dm < \epsilon

This property is called absolute continuity of the integral.

Proof of Theorem 3.6:

Proof:

Since $f \in L^1$ , we have $\int |f| < \infty$ . For $\epsilon > 0$ , choose a simple function $\varphi$ such that $0 \leq \varphi \leq |f|$ and $\int (|f| - \varphi) < \epsilon/2$ .

Let $M = \sup \varphi$ . Then for any measurable set $E$ ,

\int_E |f| = \int_E (|f| - \varphi) + \int_E \varphi \leq \frac{\epsilon}{2} + M \cdot m(E)

Choose $\delta = \frac{\epsilon}{2M}$ . Then if $m(E) < \delta$ , we have

\int_E |f| < \frac{\epsilon}{2} + M \cdot \frac{\epsilon}{2M} = \epsilon

∎

Example 3.3: Computing Lebesgue Integral of a Non-Simple Function

Problem: Compute $\int_0^1 x^2 \, dm$ using the definition of the Lebesgue integral.

Solution:

We approximate $f(x) = x^2$ by simple functions. For each $n$ , define

\varphi_n(x) = \sum_{k=0}^{n-1} \left(\frac{k}{n}\right)^2 \chi_{[k/n, (k+1)/n)}(x)

Then $0 \leq \varphi_n \leq f$ and

\int \varphi_n = \sum_{k=0}^{n-1} \left(\frac{k}{n}\right)^2 \cdot \frac{1}{n} = \frac{1}{n^3} \sum_{k=0}^{n-1} k^2 = \frac{1}{n^3} \cdot \frac{(n-1)n(2n-1)}{6}

As $n \to \infty$ , this tends to $\frac{1}{3}$ .

By the monotone convergence theorem, $\int_0^1 x^2 \, dm = \frac{1}{3}$ , which agrees with the Riemann integral.

Example 3.4: Comparison with Riemann Integral: A Specific Function

Problem: Consider the function $f(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \cap [0,1] \\ 0 & \text{if } x \in [0,1] \setminus \mathbb{Q} \end{cases}$ . Compare its Riemann and Lebesgue integrals.

Solution:

Riemann integral: This function is discontinuous at every point, so it is not Riemann integrable. The upper and lower Riemann sums do not converge to the same value.

Lebesgue integral: Since $\mathbb{Q} \cap [0,1]$ is countable, it has measure zero. Therefore, $f = 0$ almost everywhere, so

\int_0^1 f \, dm = 0

This example demonstrates that the Lebesgue integral can integrate functions that are not Riemann integrable, and it gives the "correct" answer (0) by ignoring the null set of discontinuities.

Example 3.5: Using Monotone Convergence Theorem to Compute a Limit Integral

Problem: Compute $\lim_{n \to \infty} \int_0^1 \frac{nx}{1 + n^2x^2} \, dm$ .

Solution:

Define $f_n(x) = \frac{nx}{1 + n^2x^2}$ for $x \in [0,1]$ .

For $x = 0$ , $f_n(0) = 0$ for all $n$ .

For $x > 0$ , we have $f_n(x) \to 0$ as $n \to \infty$ (since the denominator grows faster than the numerator).

However, the sequence is not monotone. Instead, we can use the dominated convergence theorem (to be covered later) or compute directly:

Using the substitution $u = nx$ , we get

\int_0^1 f_n(x) \, dx = \int_0^n \frac{u}{1 + u^2} \cdot \frac{1}{n} \, du = \frac{1}{2n} \ln(1 + n^2)

As $n \to \infty$ , this tends to 0. Therefore, $\lim_{n \to \infty} \int_0^1 f_n \, dm = 0$ .

Note that $f_n \to 0$ pointwise, but the convergence is not uniform. The dominated convergence theorem (covered in the next course) provides a more elegant solution.

Example 3.6: Constructing Functions in L1

Problem: Construct a function $f$ on $\mathbb{R}$ that is in $L^1$ but is unbounded.

Solution:

Define

f(x) = \begin{cases} \frac{1}{\sqrt{x}} & \text{if } x \in (0, 1] \\ 0 & \text{otherwise} \end{cases}

This function is unbounded (it tends to infinity as $x \to 0^+$ ), but

\int |f| = \int_0^1 \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x}\big|_0^1 = 2 < \infty

Therefore, $f \in L^1$ even though it is unbounded. This shows that $L^1$ functions need not be bounded; they only need to have finite integral of their absolute value.

Example 3.7: Absolutely Integrable but Not Riemann Integrable

Problem: Give an example of a function that is Lebesgue integrable (absolutely integrable) but not Riemann integrable.

Solution:

Consider the function $f(x) = \chi_{C}(x)$ on $[0,1]$ , where $C$ is the Cantor set.

The Cantor set has measure zero (as shown in Example 1.4), so $f = 0$ almost everywhere. Therefore,

\int_0^1 f \, dm = 0

However, $f$ is not Riemann integrable because it is discontinuous at every point of $C$ , and $C$ is uncountable (though it has measure zero).

More generally, any function that is nonzero only on a set of measure zero is Lebesgue integrable with integral 0, but may not be Riemann integrable if it has too many discontinuities.

Corollary 3.2: Properties of the L1 Norm

The $L^1$ norm $\|f\|_1 = \int |f| \, dm$ satisfies:

$\|f\|_1 \geq 0$ with equality if and only if $f = 0$ a.e.
$\|\alpha f\|_1 = |\alpha| \|f\|_1$ for $\alpha \in \mathbb{R}$
$\|f + g\|_1 \leq \|f\|_1 + \|g\|_1$ (triangle inequality)

Together with completeness (Theorem 3.5), this makes $L^1$ a Banach space.

Corollary 3.3: Continuity of the Integral

If $f \in L^1$ and $\{E_n\}$ is a sequence of measurable sets with $E_n \to \emptyset$ (in the sense that $\chi_{E_n} \to 0$ pointwise), then

\lim_{n \to \infty} \int_{E_n} f \, dm = 0

This follows from the absolute continuity property (Theorem 3.6).

Remark 3.2: Historical Development of Lebesgue Integration

The Lebesgue integral was developed by Henri Lebesgue in his 1902 thesis, "Intégrale, longueur, aire" (Integral, length, area).

Motivation: The Riemann integral, while adequate for continuous functions, had limitations:

Many functions were not Riemann integrable (e.g., functions with too many discontinuities)
Interchanging limits and integrals required strong conditions (uniform convergence)
The fundamental theorem of calculus had restrictive hypotheses

Lebesgue's insight: Instead of partitioning the domain (as in Riemann integration), partition the range. This allows integration of a much larger class of functions and provides better convergence properties.

Impact: The Lebesgue integral revolutionized analysis, providing the foundation for:

Modern functional analysis (Lp spaces, Banach spaces, Hilbert spaces)
Probability theory (expectation as an integral)
Fourier analysis and harmonic analysis
Partial differential equations

Remark 3.3: Detailed Comparison with Riemann Integration

When do Riemann and Lebesgue integrals agree?

If a function is Riemann integrable on a bounded interval, then it is Lebesgue integrable and the integrals are equal. However, the converse is false: many Lebesgue integrable functions are not Riemann integrable.

Key differences:

Domain of integration: Riemann integral is typically defined on bounded intervals; Lebesgue integral works on any measurable set, including unbounded sets and higher dimensions.
Convergence theorems: Lebesgue integral has powerful convergence theorems (monotone, dominated) that allow interchanging limits and integrals under much weaker conditions than uniform convergence required for Riemann integrals.
Null sets: Lebesgue integral ignores null sets; two functions that differ only on a null set have the same integral. This is not true for Riemann integrals.
Completeness: L1 is complete (Riesz-Fischer theorem), making it suitable for analysis. The space of Riemann integrable functions is not complete.

When to use which: For most modern analysis, the Lebesgue integral is preferred. Riemann integrals are still useful for computational purposes and when working with smooth functions, but Lebesgue theory provides the proper foundation.

Remark 3.4: Applications in Function Space Theory

The Lebesgue integral and L1 space are fundamental in functional analysis:

Lp spaces: For $1 \leq p < \infty$ , the space $L^p$ consists of functions with $\int |f|^p < \infty$ . L1 is the case $p = 1$ . All Lp spaces are Banach spaces, with L2 being a Hilbert space.
Dual spaces: The dual space of L1 (bounded linear functionals) can be identified with $L^\infty$ (essentially bounded functions), a result we'll see in later courses.
Approximation: Simple functions and continuous functions with compact support are dense in L1, providing approximation tools.
Convolution: The convolution of two L1 functions is again in L1, with $\|f * g\|_1 \leq \|f\|_1 \|g\|_1$ .
Fourier transform: The Fourier transform is well-defined on L1, though it maps to a larger space. The theory extends to L2, where it becomes an isometry.

These applications demonstrate why L1 and the Lebesgue integral are central to modern analysis.

Example 3.8: Integral Over Unbounded Sets

Problem: Compute $\int_1^\infty \frac{1}{x^2} \, dm(x)$ .

Solution:

Define $f_n(x) = \frac{1}{x^2} \chi_{[1,n]}(x)$ . Then $\{f_n\}$ is an increasing sequence of nonnegative functions converging pointwise to $f(x) = \frac{1}{x^2} \chi_{[1,\infty)}(x)$ .

By the monotone convergence theorem,

\int_1^\infty \frac{1}{x^2} \, dx = \lim_{n \to \infty} \int_1^n \frac{1}{x^2} \, dx = \lim_{n \to \infty} \left[-\frac{1}{x}\right]_1^n = \lim_{n \to \infty} \left(1 - \frac{1}{n}\right) = 1

This demonstrates how the Lebesgue integral handles unbounded domains naturally, using convergence theorems.

Example 3.9: Integral of a Function with Singularities

Problem: Compute $\int_0^1 \frac{1}{\sqrt{x}} \, dm(x)$ .

Solution:

The function $f(x) = \frac{1}{\sqrt{x}}$ is unbounded near 0, but we can compute the integral using the monotone convergence theorem.

Define $f_n(x) = \frac{1}{\sqrt{x}} \chi_{[1/n, 1]}(x)$ . Then $\{f_n\}$ is increasing and converges to $f$ pointwise.

By the monotone convergence theorem,

\int_0^1 \frac{1}{\sqrt{x}} \, dx = \lim_{n \to \infty} \int_{1/n}^1 \frac{1}{\sqrt{x}} \, dx = \lim_{n \to \infty} \left[2\sqrt{x}\right]_{1/n}^1 = \lim_{n \to \infty} \left(2 - \frac{2}{\sqrt{n}}\right) = 2

This shows that functions can be Lebesgue integrable even if they are unbounded, as long as the integral of the absolute value is finite.

Example 3.10: Change of Variables in Lebesgue Integral

Problem: Use a change of variables to compute $\int_0^\infty e^{-x^2} x \, dx$ .

Solution:

Let $u = x^2$ , so $du = 2x \, dx$ and $x \, dx = \frac{du}{2}$ .

When $x = 0$ , $u = 0$ ; when $x \to \infty$ , $u \to \infty$ .

By the change of variables formula for Lebesgue integrals,

\int_0^\infty e^{-x^2} x \, dx = \int_0^\infty e^{-u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int_0^\infty e^{-u} \, du = \frac{1}{2}

The change of variables formula for Lebesgue integrals is more general than for Riemann integrals, as it applies to measurable functions and measurable transformations.

Theorem 3.7: Change of Variables Formula

Let $\phi: [a,b] \to [c,d]$ be a strictly increasing absolutely continuous function with $\phi(a) = c$ and $\phi(b) = d$ .

If $f \in L^1([c,d])$ , then $(f \circ \phi) \phi' \in L^1([a,b])$ and:

\int_c^d f(y) \, dy = \int_a^b f(\phi(x)) \phi'(x) \, dx

This generalizes the classical change of variables formula to Lebesgue integrals.

Proof of Theorem 3.7:

Proof:

The proof uses the fact that absolutely continuous functions map null sets to null sets, and the chain rule for derivatives of absolutely continuous functions.

For simple functions, the result follows from the definition. For general functions, approximate by simple functions and use convergence theorems. ∎

∎

Corollary 3.4: Translation Invariance of the Integral

For $f \in L^1(\mathbb{R}^n)$ and $h \in \mathbb{R}^n$ , define $\tau_h f(x) = f(x - h)$ . Then:

\int \tau_h f = \int f

This follows from the translation invariance of Lebesgue measure and the change of variables formula.

Remark 3.5: Computational Techniques for Lebesgue Integrals

In practice, computing Lebesgue integrals often uses:

Fundamental theorem of calculus: When the function is absolutely continuous, use $\int_a^b f' = f(b) - f(a)$ .
Riemann integrals: For Riemann integrable functions, the Lebesgue integral equals the Riemann integral, so standard calculus techniques apply.
Convergence theorems: Approximate by simpler functions and take limits using monotone or dominated convergence.
Symmetry: Use geometric or algebraic symmetry to simplify computations.
Change of variables: Apply the change of variables formula for appropriate transformations.
Fubini's theorem: For multiple integrals, change the order of integration when possible.

These techniques, combined with the powerful convergence theorems, make Lebesgue integration both theoretically elegant and computationally practical.

Remark 3.6: Extensions and Generalizations

The Lebesgue integral has been extended in several directions:

Abstract measure spaces: The theory works on any measure space, not just $\mathbb{R}^n$ with Lebesgue measure.
Complex-valued functions: Integrate complex functions by integrating real and imaginary parts separately.
Vector-valued functions: Integrate functions taking values in Banach spaces, important for PDE theory.
Stochastic integrals: Extensions to stochastic processes, leading to Itô calculus and stochastic differential equations.
Non-absolute integrals: The Henstock-Kurzweil integral extends Lebesgue integration to handle some non-absolutely integrable functions.

These extensions demonstrate the versatility and power of the Lebesgue integral as a foundation for modern analysis.

Remark 3.1: Key Insights

Key takeaways:

The Lebesgue integral extends naturally from simple functions to general measurable functions
Absolute integrability is the key condition for a function to be in L1
Fatou's lemma and monotone convergence provide powerful tools for exchanging limits and integrals
L1 is a complete normed vector space, making it suitable for analysis
The Lebesgue integral generalizes and improves upon the Riemann integral

Practice Quiz

Lebesgue Integration Core & L1

Questions

Correct

Accuracy

The Lebesgue integral of a simple function

\varphi = \sum_{i=1}^n c_i \chi_{E_i}

is defined as:

Easy

Not attempted

For a nonnegative measurable function

f

, the Lebesgue integral is defined as:

Medium

Not attempted

A function

f

is in

L^1

if and only if:

Easy

Not attempted

Fatou's lemma states that for a sequence of nonnegative measurable functions

\{f_n\}

Medium

Not attempted

The monotone convergence theorem requires:

Medium

Not attempted

f \in L^1

and

g

is measurable with

|g| \leq |f|

a.e., then:

Easy

Not attempted

The Lebesgue integral generalizes the Riemann integral in that:

Medium

Not attempted

f_n \to f

pointwise a.e. and

|f_n| \leq g

for some

g \in L^1

, then:

Hard

Not attempted

The

L^1

norm is defined as:

Easy

Not attempted

A function that is Lebesgue integrable but not Riemann integrable is:

Hard

Not attempted

Frequently Asked Questions

What is the main advantage of Lebesgue integration over Riemann integration?

Lebesgue integration applies to a much larger class of functions and has better convergence properties. The dominated convergence theorem and monotone convergence theorem allow us to exchange limits and integrals under much weaker conditions than in Riemann integration.

How is the Lebesgue integral defined for general functions?

For a general measurable function $f$, we write $f = f^+ - f^-$ where $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ are the positive and negative parts. The integral is defined as $\int f = \int f^+ - \int f^-$, provided at least one of these is finite. If both are finite, $f$ is absolutely integrable (in $L^1$).

What is the relationship between $L^1$ and other $L^p$ spaces?

$L^1$ is the space of absolutely integrable functions. For $p > 1$, $L^p$ consists of functions with $\int |f|^p < \infty$. All $L^p$ spaces are complete normed vector spaces (Banach spaces), with $L^2$ being a Hilbert space.

Why is Fatou's lemma important?

Fatou's lemma provides a lower bound for the integral of a limit inferior, which is crucial for proving convergence theorems. It's often used as a stepping stone to prove the monotone and dominated convergence theorems.

Can we always exchange limits and integrals?

No, not in general. However, the monotone convergence theorem (for increasing sequences of nonnegative functions) and dominated convergence theorem (for sequences dominated by an $L^1$ function) provide conditions under which this exchange is valid.

What does it mean for $L^1$ to be complete?

Completeness means that every Cauchy sequence in $L^1$ converges to a function in $L^1$. This is the Riesz-Fischer theorem and is crucial for many applications in analysis, as it ensures that limits of integrable functions remain integrable.

How do we compute Lebesgue integrals in practice?

For many practical purposes, we use the fundamental theorem of calculus when the function is sufficiently regular. For more general functions, we approximate by simple functions, use convergence theorems, or relate to Riemann integrals when applicable.