Generalising Integration

In this post, we revisit Riemann integration from a generalisable lens, motivating the study of Lebesgue integration and some measure-theoretic language connected with it.

Recall the criterion required for a bounded function $f : [a, b] \to \mathbb R$ to be Riemann-integrable, with its details expanded for reasons that will become apparent.

Theorem 1. Let $f : [a, b] \to \mathbb R$ be bounded. Then $f$ is Riemann-integrable if and only if there exists a partition $P \subseteq [a, b]$ such that

$\displaystyle \sum_{i=1}^n M_i(f, P) \Delta x_i - \sum_{i=1}^n m_i(f, P) \Delta x_i < \epsilon.$

Let’s zoom-in on the quantity $\displaystyle \sum_{i=1}^n m_i(f, P) \Delta x_i$ . If we consider this quantity geometrically, we are adding rectangles with width $\Delta x_i = x_i - x_{i-1}$ and height $m_i(f, P)$ for each $i$ . The total area of the rectangles will be a lower bound on the area under $y = f(x)$ , and when $f$ is Riemann-integrable, its supremum $\mathcal L_a^b(f)$ is defined to be the integral $\displaystyle \int_a^b f$ .

Now, if instead we considered the total area $\displaystyle \sum_{i=1}^n m_i(f, P) \Delta x_i$ as an area under some function, we would get the function $\phi : [a, b] \to \mathbb R$ defined by

$\displaystyle \phi = \sum_{i=1}^n m_i(f, P) \cdot \mathbb I_{[x_{i-1}, x_i)},\quad \phi(b) = m_n(f,P).$

There are a total of $n$ steps, and at step $i$ , the height of the step function in the $i$ -th interval is $m_i(f, P)$ . This motivates the definition of a simple function.

Definition 1. Let $\phi : [a, b] \to \mathbb R$ be a non-negative bounded function. We say that $\phi$ is simple if there exists $\{a_1,\dots,a_n\}$ and a partition $P \subseteq [a, b]$ such that

$\displaystyle \phi = \sum_{i=1}^n a_i \cdot \mathbb I_{[x_{i-1}, x_i)},\quad \phi(b) = a_n.$

We remark that the $[x_{i-1},x_i)$ are all mutually disjoint, and

$\displaystyle [a, b] = \bigcup_{i=1}^n [x_{i-1},x_i) \cup \{b\}.$

We also remark that for each $1\leq i <n$ , $\phi^{-1}(\{a_i\}) = [x_{i-1}, x_i)$ , and $\phi^{-1}(\{a_n\}) = [x_{n-1}, x_n]$ . Using these notations,

$\displaystyle \phi = \sum_{i=1}^n a_i \cdot \mathbb I_{\phi^{-1}(\{a_i\})},$

where we denote $\{a_i\} = a_i$ for brevity.

The area under this simple function, as the name suggests, is rather simple too, as a sum of areas of rectangles. Each rectangle has base $x_i - x_{i-1}$ , which is the length of the interval $[x_{i-1},x_i)$ .

Letting $\lambda$ denote a length function (i.e. a measure), we denote

$\lambda(\phi^{-1}(\{a_n\})) = x_i - x_{i-1} = \Delta x_i.$

With this notation, we can define the Lebesgue integral of a simple function.

Definition 2. For any $K \subseteq [a, b]$ such that $\lambda(K)$ exists, we define

$\displaystyle \int_K 1\, \mathrm d\lambda \equiv \int_{[a, b]} \mathbb I_{K}\, \mathrm d\lambda := \lambda(K).$

Let $\phi : [a, b] \to \mathbb R$ be a simple function with range $\{a_1,\dots,a_n\}$ . The Lebesgue integral of $\phi$ is then defined via linearity to equal

$\displaystyle \begin{aligned} \int_{[a, b]} \phi\, \mathrm d\lambda &= \int_{[a, b]} \left(\sum_{i=1}^n a_i \cdot \mathbb I_{\phi^{-1}(\{a_i\})}\right)\, \mathrm d\lambda \\ &:= \sum_{i=1}^n a_i \cdot \int_{[a, b]} \mathbb I_{\phi^{-1}(\{a_i\})}\, \mathrm d\lambda \\ &= \sum_{i=1}^n a_i \cdot \lambda(\phi^{-1}(\{a_i\}))\, \mathrm d\lambda.\end{aligned}$

We call $\lambda$ the Lebesgue measure, so that sets $K$ where $\lambda(K)$ is well-defined are called Lebesgue-measurable sets, in the following manner.

Theorem 2. There exists a collection $\mathcal F$ of subsets of $\mathbb R$ containing subsets of the form $[a, b]$ and a function $\lambda : \mathcal F \to \mathbb R_{\geq 0} \cup \{+\infty\}$ that satisfies the following properties:

$\emptyset \in \mathcal F, \mathbb R \in \mathcal F$ ,
For any $K \in \mathcal F$ , $\mathbb R \backslash K \in \mathcal F$ ,
For disjoint $K_1,K_2,\dots \in \mathcal F$ , $\displaystyle \bigcup_{i=1}^\infty K_i \in \mathcal F$ .

We call $\mathcal F$ the Lebesgue $\sigma$ -algebra whose elements are called Lebesgue measurable sets (more on this name later on), and $(\mathbb R, \mathcal F)$ is called a measurable space. Furthermore,

$\lambda(\emptyset) = 0$ ,
For any $K \in \mathcal F$ , $\lambda(K) \geq 0$ ,
For disjoint $K_1,K_2,\dots \in \mathcal F$ , $\displaystyle \lambda\left(\bigcup_{i=1}^\infty K_i \right) = \sum_{i=1}^\infty \lambda(K_i)$ , where the right-hand side may be $+ \infty$ . This is known as countable additivity.
For any $a \leq b$ , $\lambda([a, b]) = b-a$ .

We call $\lambda$ the Lebesgue measure on $\mathbb R$ , and $(\mathbb R, \mathcal F, \lambda)$ the corresponding measure space. The definitions are stipulated so that anything measurable is, quite literally measurable, that is, agrees with our definition of a measure.

Proof. Omitted. Or relegated to a discussion on measure theory.

Furthermore, functions like $\phi$ where integration makes sense are called Lebesgue-measurable functions. If their integrals are finite, we say they are Lebesgue-integrable. We formalise these ideas below.

Definition 3. A function $f :\mathbb R \to \mathbb R$ is Lebesgue-measurable if for any $a \in \mathbb R$ , $f^{-1}((a, \infty)) \in \mathcal F$ . It can be shown that $f^{-1}(\{a\}) \in \mathcal F$ .

For any $K \subseteq \mathcal F$ , $\mathbb I_K$ is Lebesgue-measurable if and only if $K$ is Lebesgue-measurable. Hence, if $f$ is simple, then $f$ is Lebesgue-measurable if and only if $f^{-1}(\{a_i\})$ is Lebesgue-measurable for each $i$ . In this case, we can proceed with the definitions in Definition 2.

If $f \geq 0$ , we define

$\displaystyle \int_{[a, b]}f\,\mathrm d\lambda := \sup_{0 \leq \phi \leq f} \int_{[a, b]} \phi\, \mathrm d\lambda,$

where the supremum is taken over simple functions $\phi$ that satisfy $0 \leq \phi \leq f$ .

To study these objects in more detail and verify that they do agree with our standard notions of Riemann integrability shoots us into a whole new stratosphere of study called measure theory, which lies far beyond the scope of introductory real analysis. Nevertheless, we can restate the crucial Riemann integrability criterion using Lebesgue notions.

Theorem 3. Let $f : [a, b] \to \mathbb R$ be bounded. For any partition $P =\{x_0,x_1,\dots,x_n\}$ , make the following definitions