Simple Function

From Optimal Transport Wiki
Jump to navigation Jump to search

The simplest functions you will ever integrate, hence the name.


Let be a measure space. A measurable function is a simple function[1] if is a finite subset of . The standard representation[1] for a simple function is given by


where is the indicator function on the disjoint sets that partition , where .


Consider the functions determined by [2] and . The function is a simple function and can be expressed in the following manner:

The last of these representations is the standard representation of . The function is not a simple function as .

Integration of Simple Functions

These functions earn their name from the simplicity in which their integrals are defined[3]. Let be the space of all measurable functions from to Then

where by convention, we let . Note that is equivalent to and that some arguments may be omitted when there is no confusion.

Furthermore, for any , we define

Properties of Simple Functions

Given simple functions , the following are true[3]:

  • if ;
  • ;
  • if , then ;
  • the function is a measure on .


Let and be simple functions with their corresponding standard representations.

We show the first claim. Suppose . Then , implying . Similarly, . Thus, the first statement holds for this case.

Suppose . Then


Next, we show the second statement. Notice that we can rewrite and as unions of disjoint sets as follows



which by countable additivity,

It is worth noting that this may not be the standard representation for the integral of .

As for the third statement, if , then whenever , it must be that , implying that

Finally, we show the last statement. Define . Now we show satisfies all the measure properties. Notice that is a nonnegative function on . Then compute

Consider a disjoint sequence of sets and let be its union. Then

which by countable additivity is equal to


  1. 1.0 1.1 Craig, Katy. MATH 201A Lecture 11. UC Santa Barbara, Fall 2020.
  2. Craig, Katy. MATH 201A Lecture 12. UC Santa Barbara, Fall 2020.
  3. 3.0 3.1 Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.
  4. Craig, Katy. MATH 201A Lectures 12-13. UC Santa Barbara, Fall 2020.