Measure Theory/Properties of Simple Integrals

Properties of Simple Integrals

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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   then the simple integrals satisfy  . This will allow us to prove that if two measurable function satisfy   then the length-measure integrals satisfy  , 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,  , in this lesson,

  • there is some   such that  
  • for all  .

Linearity

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

Let   be any two simple functions. (Assume that   has finite measure and outside of G, both of these functions are identically zero.)

Exercise 1. Constrained Simple Integrals Exist

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Show that   is a finite real number.

Exercise 2. Linearity of Simple Integration

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Show, using an earlier exercise about the sum of simple functions, that  .

Also show that if   then  .

Infer linearity:

 

Inequality Preserving

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With   as above, suppose further that  .

Exercise 3. Prove Inequality Preserving

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Prove that  .

Hint: From   infer  . Now use this to argue that   and from there, infer the desired result by appealing to linearity.