Home/Chapter 13

Appendix A: Integration and Some Useful Properties

Measurable spaces, Borel sigma-fields, Lebesgue integration, Fatou's lemma, monotone and dominated convergence theorems, differentiation under the integral, and Fubini's theorem.

This appendix collects the key definitions and results from measure theory and integration that are used throughout the course. These results underpin the rigorous treatment of $L_p$ spaces, Fourier transforms, and convergence theorems.

A.1 Measurable Space

DefinitionSigma-field (Sigma-algebra)

Let $\mathbb{X}$ be a collection of points. Let $\mathcal{F}$ be a collection of subsets of $\mathbb{X}$ with the following properties such that $\mathcal{F}$ is a $\sigma$ -field (also called a $\sigma$ -algebra), that is:

$\mathbb{X} \in \mathcal{F}$
If $A \in \mathcal{F}$ , then $\mathbb{X} \setminus A \in \mathcal{F}$
If $A_k \in \mathcal{F}$ , $k = 1, 2, 3, \ldots$ , then $\bigcup_{k=1}^{\infty} A_k \in \mathcal{F}$ (that is, the collection is closed under countably many unions).

Remark.

Intuition: A $\sigma$ -field tells us which subsets of a space we are "allowed" to measure. It must include the whole space, be closed under complements, and be closed under countable unions. This structure ensures that the operations we perform on measurable sets (unions, intersections, complements) always produce measurable sets.

By De Morgan's laws and set properties, the collection also has to be closed under countable intersections.

If the third item above holds for only finitely many unions or intersections, then the collection of subsets is said to be a field or algebra.

With the above, $(\mathbb{X}, \mathcal{F})$ is termed a measurable space (that is, we can associate a measure to this space, which we will discuss shortly). For example, the full power-set of any set is a $\sigma$ -field.

A $\sigma$ -field $\mathcal{J}$ is generated by a collection of sets $\mathcal{A}$ , if $\mathcal{J}$ is the smallest $\sigma$ -field containing the sets in $\mathcal{A}$ , and in this case, we write $\mathcal{J} = \sigma(\mathcal{A})$ .

A.1.1 Borel $\sigma$ -field

DefinitionBorel Sigma-field

An important class of $\sigma$ -fields is the Borel $\sigma$ -field on a metric (or more generally, topological) space. Such a $\sigma$ -field is the one which is generated by open sets. The term open naturally depends on the space being considered.

Remark.

Intuition: The Borel $\sigma$ -field is the "natural" collection of measurable sets on $\mathbb{R}$ (or any metric space). It is generated by open sets, which means it contains all open intervals, closed intervals, countable unions and intersections of these, and so on. Essentially all sets you encounter in practice are Borel sets.

For this course, we will mainly consider spaces which are complete, separable and metric spaces (such as the space of real numbers $\mathbb{R}$ , or countable sets). Recall that in a metric space with metric $d$ , a set $U$ is open if for every $x \in U$ , there exists some $\epsilon > 0$ such that $\{y : d(x, y) < \epsilon\} \subset U$ . We note also that the empty set is a special open set.

The Borel $\sigma$ -field on $\mathbb{R}$ is then the one generated by sets of the form $(a, b) \subset \mathbb{R}$ , that is, open intervals. We will denote the Borel $\sigma$ -field on a space $\mathbb{X}$ as $\mathcal{B}(\mathbb{X})$ .

A.1.2 Measurable Function

DefinitionMeasurable Function

If $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$ and $(\mathbb{Y}, \mathcal{B}(\mathbb{Y}))$ are measurable spaces, we say a mapping from $h : \mathbb{X} \to \mathbb{Y}$ is measurable if

$h^{-1}(B) = \{x : h(x) \in B\} \in \mathcal{B}(\mathbb{X}), \qquad \forall B \in \mathcal{B}(\mathbb{Y})$

TheoremChecking Measurability

To show that a function is measurable, it is sufficient to check the measurability of the inverses of sets that generate the $\sigma$ -algebra on the image space.

Remark.

Intuition: You do not need to check measurability for every Borel set in the range -- it suffices to check it for a generating collection (such as open intervals for real-valued functions). This dramatically simplifies verifying measurability in practice.

Therefore, for Borel measurability, it suffices to check the measurability of the inverse images of open sets. Furthermore, for real valued functions, to check the measurability of the inverse images of open sets, it suffices to check the measurability of the inverse images sets of the form $\{(-\infty, a], a \in \mathbb{R}\}$ , $\{(-\infty, a), a \in \mathbb{R}\}$ , $\{(a, \infty), a \in \mathbb{R}\}$ or $\{[a, -\infty), a \in \mathbb{R}\}$ , since each of these generate the Borel $\sigma$ -field on $\mathbb{R}$ . In fact, here we can restrict $a$ to be $\mathbb{Q}$ -valued, where $\mathbb{Q}$ is the set of rational numbers.

A.1.3 Measure

DefinitionMeasure

A positive measure $\mu$ on $(\mathbb{X}, \mathcal{B}(\mathbb{X}))$ is a map from $\mathcal{B}(\mathbb{X})$ to $[0, \infty]$ which is countably additive such that for $A_k \in \mathcal{B}(\mathbb{X})$ and $A_k \cap A_j = \emptyset$ :

$\mu\left(\cup_{k=1}^{\infty} A_k\right) = \sum_{k=1}^{\infty} \mu(A_k).$

DefinitionProbability Measure

$\mu$ is a probability measure if it is positive and $\mu(\mathbb{X}) = 1$ .

DefinitionFinite and Sigma-finite Measures

A measure $\mu$ is finite if $\mu(\mathbb{X}) < \infty$ , and $\sigma$ -finite if there exist a collection of subsets such that $X = \cup_{k=1}^{\infty} A_k$ with $\mu(A_k) < \infty$ for all $k$ .

Remark.

Intuition: A measure assigns a "size" to sets in a consistent way. Countable additivity ensures that the measure of a disjoint union equals the sum of the measures of the pieces. The Lebesgue measure on $\mathbb{R}$ , which assigns to each interval $(a,b)$ its length $b - a$ , is the prototypical example used throughout this course.

On the real line $\mathbb{R}$ , the Lebesgue measure is defined on the Borel $\sigma$ -field (in fact on a somewhat larger field obtained through adding all subsets of Borel sets of measure zero: this is known as completion of a $\sigma$ -field) such that for $A = (a, b)$ , $\mu(A) = b - a$ . Borel field of subsets is a subset of Lebesgue measurable sets, that is there exist Lebesgue measurable sets which are not Borel sets. There exist Lebesgue measurable sets of measure zero which contain uncountably many elements; an example is the Cantor set.

A.1.4 The Extension Theorem

TheoremThe Extension Theorem (Caratheodory)

Let $\mathcal{M}$ be an algebra over $\mathbb{X}$ , and suppose that there exists a map (called a pre-measure) $P : \mathcal{M} \to \mathbb{R}_+$ so that for any (possibly countably infinitely many) pairwise disjoint sets $A_n \in \mathcal{M}$ , if the countable union $\cup_n A_n \in \mathcal{M}$ , then $P(\cup_n A_n) = \sum_n P(A_n)$ . Suppose also that there exists a countable collection of sets $B_n$ with $\mathbb{X} = \cup_n B_n$ , each with $P(B_n) < \infty$ (that is $P$ is $\sigma$ -finite). Then, there exists a unique measure $P'$ on the $\sigma$ -field generated by $\mathcal{M}$ , $\sigma(\mathcal{M})$ , which is consistent with $P$ on $\mathcal{M}$ .

Remark.

Intuition: The Extension Theorem says that if you define a consistent "pre-measure" on a simple collection of sets (like intervals), there is a unique way to extend it to the full Borel $\sigma$ -field. This is how the Lebesgue measure is constructed: we define $\mu(a, b) = b - a$ on intervals and extend uniquely to all Borel sets.

The above is useful since, when one states that two measures are equal it suffices to check if they are equal on the set of sets which generate the $\sigma$ -algebra, and not necessarily on the entire $\sigma$ -field.

We can construct the Lebesgue measure on $\mathcal{B}(\mathbb{R})$ by defining it on finitely many unions and intersections of intervals of the form $(a, b)$ , $[a, b)$ , $(a, b]$ and $[a, b]$ , and the empty set, thus forming a field, and extending this to the Borel $\sigma$ -field. Thus, the relation $\mu(a, b) = b - a$ for $b > a$ is sufficient to define the Lebesgue measure.

A.1.5 Integration

The Lebesgue integral is constructed as follows. A simple function is a finite linear combination of indicator functions:

$f(x) = \sum_{i=1}^{n} a_i 1_{\{x \in A_i\}}$

where $A_i$ are measurable sets. The integral of a simple function is defined as:

$\int f\,d\mu = \sum_{i=1}^{n} a_i \mu(A_i).$

For a non-negative measurable function $f$ , the integral is defined as:

$\int f\,d\mu = \sup\left\{\int g\,d\mu : g \text{ simple}, 0 \leq g \leq f\right\}.$

For a general measurable function, write $f = f^+ - f^-$ where $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ , and define $\int f\,d\mu = \int f^+\,d\mu - \int f^-\,d\mu$ provided at least one of these is finite.

A.1.6 Fatou's Lemma, the Monotone Convergence Theorem and the Dominated Convergence Theorem

The following three results are fundamental convergence theorems for Lebesgue integration, used extensively throughout the course.

TheoremFatou's Lemma

Let $\{f_n\}$ be a sequence of non-negative measurable functions. Then,

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

Remark.

Intuition: Fatou's Lemma says that the integral of the limit is at most the limit of the integrals. Mass can "escape to infinity" or "concentrate into a point" during a limiting process, so the limit of the integrals might be strictly larger. This lemma provides a one-sided bound that always holds for non-negative functions.

TheoremMonotone Convergence Theorem

Let $\{f_n\}$ be a sequence of non-negative measurable functions such that $f_n \leq f_{n+1}$ for all $n$ (pointwise). Then,

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

Remark.

Intuition: For monotonically increasing non-negative functions, the limit and the integral can be interchanged freely. The monotonicity prevents mass from escaping, which is why the inequality in Fatou's Lemma becomes an equality here.

TheoremDominated Convergence Theorem

Let $\{f_n\}$ be a sequence of measurable functions such that $f_n \to f$ pointwise (or almost everywhere). If there exists an integrable function $g$ (that is, $\int |g|\,d\mu < \infty$ ) such that $|f_n| \leq g$ for all $n$ , then

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

Remark.

Intuition: The Dominated Convergence Theorem is the most commonly used convergence theorem in this course. If a sequence of functions converges pointwise and is uniformly bounded by an integrable function, then limit and integral commute. The "dominating" function $g$ prevents mass from escaping to infinity. This theorem is used repeatedly in proofs involving Fourier transforms, approximate identities, and $L_p$ convergence.

A.2 Differentiation under an Integral

TheoremDifferentiation under the Integral Sign (Leibniz Rule)

Let $f(x, t)$ be a function such that both $f$ and its partial derivative $\frac{\partial f}{\partial x}$ are continuous in both $x$ and $t$ , and suppose there exists an integrable function $g(t)$ such that $\left|\frac{\partial f}{\partial x}(x, t)\right| \leq g(t)$ for all $x$ in a neighborhood and all $t$ . Then,

$\frac{d}{dx} \int f(x, t)\,dt = \int \frac{\partial f}{\partial x}(x, t)\,dt.$

Remark.

Intuition: This result allows us to "push the derivative inside the integral." It is used extensively in computing derivatives of Laplace and Fourier transforms (e.g., the differentiation property in the relevant section). The key condition is the existence of an integrable dominating function for the partial derivative, which allows an application of the Dominated Convergence Theorem.

A.3 Fubini's Theorem (also Fubini-Tonelli's Theorem)

TheoremFubini's Theorem

Let $(\mathbb{X}, \mathcal{F}_X, \mu)$ and $(\mathbb{Y}, \mathcal{F}_Y, \nu)$ be two $\sigma$ -finite measure spaces. Let $f : \mathbb{X} \times \mathbb{Y} \to \mathbb{R}$ be measurable with respect to the product $\sigma$ -field $\mathcal{F}_X \otimes \mathcal{F}_Y$ .

(Tonelli) If $f \geq 0$ , then

$\int_{\mathbb{X}} \left(\int_{\mathbb{Y}} f(x, y)\,d\nu(y)\right) d\mu(x) = \int_{\mathbb{Y}} \left(\int_{\mathbb{X}} f(x, y)\,d\mu(x)\right) d\nu(y) = \int_{\mathbb{X} \times \mathbb{Y}} f\,d(\mu \times \nu).$

(Fubini) If $\int_{\mathbb{X} \times \mathbb{Y}} |f|\,d(\mu \times \nu) < \infty$ , then the same equality holds and all three integrals are finite.

Remark.

Intuition: Fubini's Theorem says that for "well-behaved" functions (either non-negative, or absolutely integrable), the order of integration does not matter. This is the rigorous justification for swapping the order of double integrals, which is used throughout the course in convolution calculations, Parseval's theorem, and transfer function computations.

A.1 Measurable Space

A.1.1 Borel σ\sigmaσ-field

A.1.2 Measurable Function

A.1.3 Measure

A.1.4 The Extension Theorem

A.1.5 Integration

A.1.6 Fatou's Lemma, the Monotone Convergence Theorem and the Dominated Convergence Theorem

A.2 Differentiation under an Integral

A.3 Fubini's Theorem (also Fubini-Tonelli's Theorem)

A.1.1 Borel $\sigma$ -field