Some notes on the general theory of integration (WIP).

# Fundamentals

## Daniell Integral Axioms

The Daniell axiomatization of the integral is built on three objects: a set $\Omega$, a collection $H$ of "integrable" real-valued functions of the form $h:\Omega\to\bf R$, and an **integral function** $\int:H\to\bf R$. As convention, we write $\int h$ in place of $\int (h)$ for $h\in H$.

There is no restriction on the set $\Omega$, but $H$ must satisfy the following closure properties:

- $H$ is a vector space. That is, $f+\lambda g\in H$ whenever $f,g\in H$ and $\lambda\in\bf R$ .
- $\left|f\right|\in H$ whenever $f\in H$.

$H$ is sometimes said to be a **vector lattice** of functions on $\Omega$.

**Proposition.** $f \land g = \min (f,g)\in H$ and $f \lor g = \max (f,g)\in H$ whenever $f,g\in H$.

*Proof.* Clearly, $f+g+|f-g| = 2 \max (f,g)$ and $\min (f, g) = - \max(-f,
-g)$. ♠️

In addition to the closure properties of $H$ , the following restrictions must be placed on the integral function $\int$:

- (
*Linearity*) $\int f + \lambda g = \int f + \lambda \int g$ whenever $f,g\in H$ and $\lambda\in\bf R$ That is to say, $\int$ acts as a linear functional over $H$. - (
*Monotonicity*) $\int f \geq 0$ whenever $f\in H$ and $f \geq 0$. Equivalently, $\int f \geq \int g$ for any $g,f\in H$ where $f \geq g$. - (
*Monotone Continuity*) If $f_1, f_2, \ldots$ is a monotonic sequence of functions converting to zero, then the corresponding sequence of integrals $\int f_1, \int f_2, \ldots$ also converges to zero.

## Monotone Convergence

For a given space $H$ of integrable functions, one can define a superset $\bar H$ such that a compatable integral $\int$ on $H$ may be extended in a compatible way to an integral $\bar{\int}$ on $\bar H$.

We define $\bar H$ to consist of all real-valued functions $f:\Omega\to\bf R$ such that there exists a sequence $f_n$ of functions in $H$ that converge monotonically to $f$ and that the sequence of integrals $\int f_n$ converge to some finite value.

The function $f$ may or may not be in $H$. But if it where, the monotone continuity axiom would imply that $\int f = \lim \int f_n$. This motivates defining the "extended integral" $\bar{\int}$ on $\bar H$ as follows: $\bar{\int} f = \lim \int f_n.$

Note that this definition is only well-defined if the above limit were independent of the particular approximating sequence $f_n$, since $f$ may have multiple of such approximating sequences. We establish such a fact with the following proposition:

**Proposition.** If two monotonic sequences of integrable functions have the same limit, then their corresponding sequences of integrals also share a limit (possibly $\pm\infty$).

## Lebesque Integrals

An integral obtained by the above "monotonic closure" procedure is called a **Lebesque integral**.