Measure Theory/Properties of Simple Integrals

Properties of Simple Integrals

The properties of simple integrals are not truly interesting for their own sake -- as far I know, anyway.

But rather they are tools that we will be happy to have, when we prove the corresponding facts for length-integrals. For example, we will prove that if two simple functions satisfy $\varphi \leq \psi$ then the simple integrals satisfy $\int \varphi \leq \int \psi$. This will allow us to prove that if two measurable function satisfy $f\leq g$ then the length-measure integrals satisfy $\int f\leq \int g$, and this is the result that is truly of interest.

Throughout this lesson we will assume that whenever a simple function is given, it is given in canonical form.

We will also assume that the functions are non-zero on a set of finite measure. That is to say, we will assume for every simple function, $\psi$, in this lesson,

• there is some $E\in {\mathcal {M}}$ such that $\lambda (E)<\infty$
• for all $x\in E^{c},\quad \psi (x)=0$.

Linearity

We will prove that simple integration distributes over sums and scalar multiples.

Let $\varphi =\sum _{i=1}^{m}c_{i}\mathbf {1} _{E_{i}},{\text{ and }}\psi =\sum _{i=1}^{n}d_{i}\mathbf {1} _{F_{i}}$ be any two simple functions. (Assume that $G\in {\mathcal {M}}$ has finite measure and outside of G, both of these functions are identically zero.)

Exercise 1. Constrained Simple Integrals Exist

Show that $\int \varphi$ is a finite real number.

Exercise 2. Linearity of Simple Integration

Show, using an earlier exercise about the sum of simple functions, that $\int (\varphi +\psi )=\int \varphi +\int \psi$.

Also show that if $c\in {\mathbb {R}}$ then $\int c\varphi =c\int \varphi$.

Infer linearity:

$\forall c,d\in {\mathbb {R}},\quad \int (cf+dg)=c\int f+d\int g$

Inequality Preserving

With $\varphi ,\psi$ as above, suppose further that $\varphi \leq \psi$.

Exercise 3. Prove Inequality Preserving

Prove that $\int \varphi \leq \int \psi$.

Hint: From $\varphi \leq \psi$ infer $0\leq \psi -\varphi$. Now use this to argue that $0\leq \int (\psi -\varphi )$ and from there, infer the desired result by appealing to linearity.