Basic Notions of Lebesgue Integration

The main idea is to define in a simple and fast way the Lebesgue integral and the Lebesgue spaces, since they will be used in this site, both in the study of the universal approximation theorem and in the study of partial differential equations.

For ease of writing, the following sets will be introduced, $\overline{\mathbb{R}} = \mathbb{R}\cup\{-\infty, +\infty\}$ and $\overline{\mathbb{R}}^+ =\mathbb{R}^+\cup\{+\infty\}$.

Some Definitions

A $\pmb{\sigma\text{-algebra}}$ is a collection $\Sigma$ of subsets of $\mathbb{R}^n$ with the properties:

1. The empty set $\empty$ and $\mathbb{R}^n$ $\in$ $\Sigma$;
2. If $A \in \Sigma$, then its complement $A^c \in \Sigma$ ;
3. $A_j\in\Sigma$, $j=1,2,...$, then $\bigcup_{j=1}^{\infty}A_j\in\Sigma$.

A set $A\in\Sigma$ is said to be measurable.

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Let $\Sigma$ be a $\sigma\text{-algebra}$. A function $\mu:\Sigma\to\overline{\mathbb{R}}^+$ is a measure if it has the properties:

1. $\mu(\empty)=0$;
2. $\mu$ is a countably additive, that is, if $A_j\in\Sigma$, $j=1,2,...$, are pairwise disjoint sets then

$$\mu\left(\bigcup_{j=1}^{\infty}A_j\right)=\sum_{j=1}^{\infty}\mu(A_j).$$

Let $\Sigma$ be a $\sigma\text{-algebra}$ and a measure $\mu$ on $\Sigma$ such that $A_j=\{x\in\mathbb{R}^n:a_i\leq x_i\leq b_i, i=1,2,..., n\}$ and

$$\mu(A_j)=\prod_{i=1}^n(b_i-a_i).$$

The elements of $\Sigma$ are called Lebesgue measurable subsets of $\mathbb{R}^n$ , and $\mu$ is called the Lebesgue measure in $\mathbb{R}^n$

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Let $A\sub\mathbb{R}^n$. A function $f:A\to\overline{\mathbb{R}}$ is said measurable if the set

$$\{x\in A: f(x)>\alpha\}$$

is measurable for every $\alpha\in\mathbb{R}$.

If $f$ is measurable then the functions $|f|$ are measurable.

Construction of the integral

Let $A\sub\mathbb{R}^n$. In the following sequence of elements we describe the construction of the integral of appropriate functions $f: A \to \overline{\mathbb{R}}.$

For any subset $A_j\subset\mathbb{R}^n$. The characteristic function $\chi_{A_j}$ of $A_j$ is defined by,

$$\chi_{A_j}(x)= \begin{cases} 1 & \ x \in A_j, \\ 0 & \ x \notin A_j. \end{cases}$$

A function $\phi:A\to\mathbb{R}$ is simple if its range is a finite set of real numbers, this is $\phi(x)\in \{ \alpha_1,...,\alpha_m \}$ , then

$$\phi = \sum^k_{j=1}\alpha_j\chi_{A_j},$$

where $A_j=\{x\in A: \phi(x)=\alpha_j \}$, and $\phi$ is measurable if and only if $A_j$ are measurable, $j=1,...,k.$

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For simple function $\phi$, where $A_j\sub A$, $A_j$ measurable, $j=1,...,k.$ We define

$$\int_A\phi(x)d\mu = \sum^k_{j=1}\alpha_j\mu(A_j).$$

If $f$ is measurable and non-negative and let the integral of $f$ be defined by

$$\int_A f d\mu = \sup \left\{ \int_A\phi d\mu: \phi \ \text{is simple and } 0\leq\phi\leq f \right\}.$$

If $f$ is measurable and $\int_A |f| d\mu<\infty$, then $f$ can be seen as $f^+-f⁻$, where $f^+=\max(f,0)$ and $f^-=-\min(f,0)$ are both measurable and non-negative and

$$\int_A f d\mu = \int_A f^+d\mu - \int_A f^- d\mu,$$

and $\mu$ is the Lebesgue measure then it is said that $f$ is Lebesgue Integrable.

The Lebesgue Space

Definition $L^1(\Omega)$

Let equivalence relation $\sim$ on all Lebesgue Integrable functions of $\Omega \sub\mathbb{R}^n$ such

$$f \sim g, \quad f(x)=g(x) \ \text{almost everywhere} \ x\in\Omega.$$

The generated equivalence class is denoted $\pmb{L^1(\Omega)}$.

Definition $L^p(\Omega)$

Let $\Omega \sub\mathbb{R}^n$ and let $p\in\mathbb{N}$. $\pmb{L^p(\Omega)}$ is the class equivalence (with equivalence relation $\sim$) of all measurable functions $f:\Omega\to\mathbb{R}$ with the property

$$\int_\Omega |f|^p d\mu<\infty.$$

References

For the construction of the Lebesgue integral and the development of the Lebesgue spaces I was strongly inspired by 1 but adapted to the notation and concept used in 2, The exposition is intentionally detailed and self-contained to avoid requiring the reader to consult external references.

1. B. P. Rynne and M. A. Youngson, Linear Functional Analysis, 2nd ed. London: Springer, 2008, pp. 20-28, 34.

2. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. Amsterdam: Academic Press, 2003, pp. 13-16.