Integration

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 Ω\Omega, a collection HH of "integrable" real-valued functions of the form h:ΩRh:\Omega\to\bf R, and an integral function :HR\int:H\to\bf R. As convention, we write h\int h in place of (h)\int (h) for hHh\in H.

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

  1. HH is a vector space. That is, f+λgHf+\lambda g\in H whenever f,gHf,g\in H and λR\lambda\in\bf R .
  2. fH\left|f\right|\in H whenever fHf\in H.

HH is sometimes said to be a vector lattice of functions on Ω\Omega.

Proposition. fg=min(f,g)Hf \land g = \min (f,g)\in H and fg=max(f,g)Hf \lor g = \max (f,g)\in H whenever f,gHf,g\in H.

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

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

  1. (Linearity) f+λg=f+λg\int f + \lambda g = \int f + \lambda \int g whenever f,gHf,g\in H and λR\lambda\in\bf R That is to say, \int acts as a linear functional over HH.
  2. (Monotonicity) f0\int f \geq 0 whenever fHf\in H and f0f \geq 0. Equivalently, fg\int f \geq \int g for any g,fHg,f\in H where fgf \geq g.
  3. (Monotone Continuity) If f1,f2,f_1, f_2, \ldots is a monotonic sequence of functions converting to zero, then the corresponding sequence of integrals f1,f2,\int f_1, \int f_2, \ldots also converges to zero.

Monotone Convergence

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

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

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


Created 2019-12-11. Last updated 2020-06-22. View source.