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 $$\varphi\le\psi$$ then the simple integrals satisfy $$\int\varphi\le\int\psi$$. This will allow us to prove that if two measurable function satisfy $$f\le g$$ then the length-measure integrals satisfy $$\int f\le \int g$$, 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 $$c\in\Bbb R$$ then $$\int c\varphi = c\int \varphi$$.

Infer linearity:


 * $$\forall c,d\in\Bbb R,\quad \int(cf+dg) = c\int f+d\int g$$

Inequality Preserving
With $$\varphi,\psi$$ as above, suppose further that $$\varphi\le\psi$$.

Exercise 3. Prove Inequality Preserving
Prove that $$\int\varphi\le\int\psi$$.

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