# Measure Theory

Completion status: this resource is ~90% complete. |

Subject classification: this is a mathematics resource. |

## Design Principles

[edit | edit source]This course is designed with a few principles in mind.

### Natural Proofs

[edit | edit source]The most core design principle for this course, which I am calling "natural proofs", is

*No step, whether it's a definition, theorem, or step in a proof, should ever "come out of nowhere".*

Any abandonment of this principle, basically gives up on the job of being a *teacher* rather than just a *reference source*. Besides that, it fails to expose the actual process of discovering mathematics, which is the fundamental skill that one should learn as a mathematician.

The litmus test for whether a leap is too large, is this: Would an average student (measured however you like) feel as though they could have eventually come up with this step, given enough time and observations? If a student does not feel that the could have come up with a step, on their own, then the course instructions should break the task into smaller and more natural components. Often this can be done by making certain smaller observations or guesses, which could motivate the needed step.

But these observations and guesses should never seem inhuman: Again, after making these intermediate motivations, the author should again apply the test: "is this sufficient for the student to feel that they could have come up with it themselves, given time and observations?"

### Other Design Principles

[edit | edit source]- Exercises are embedded as part of the text, and these are as frequent and manageable as possible. As soon as something is introduced, there should be exercises on it, if at all possible. But these kinds of exercises should not discourage or impede a student -- they should just invite the student to "get their hands dirty" with these new things they've just encountered.
- Hard exercises, or results that are not essential, can be good as exercise. However, those should be kept in pages containing "Optional Exercises" in their titles.
- History and context are used "just enough" to help the reader understand what these things are and why we're doing this.

Because exercises are a part of the text, I will eventually want to make a solution guide. However, I will only try to get around to this after I've completed the main content of the text.

## References

[edit | edit source]The main body of the course is made up of lessons, and it is designed for you to primarily use the lessons.

However, it can be helpful to have condensed and organized summaries of the information.

These are linked below.

## Lesson Schedule

[edit | edit source]### Section 0: Introduction

[edit | edit source]Introduction Optional Exercises

### Section 1: Measuring Sets of Reals

[edit | edit source]Measuring Reals Optional Exercises

- Results proved in this lesson.

Theorem: There Is No Total Measure of Real Numbers Let be any real-valued set function. Then must not satisfy at least one of the properties: Length-measure, nonnegativity, translation-invariance, countable additivity.

No Total Measure Optional Exercises

- Results proved in this lesson.

Theorem: Outer-measure Is Well-defined and Nonnegative. For every the outer-measure takes a unique extended real number value. That is to say . Also this value is nonnegative, .

```
Theorem: The outer measure by countable sums.
```

Theorem: Outer-measure Is Translation Invariant. For every subset and real number the outer-measure ofAis invariant under translation byc,

Theorem: Outer-measure Is Monotonic. If then .

Theorem: Countable Sets Are Null. If is a countable set, then .

Outer Measure Optional Exercises

- Results proved in this lesson.

Theorem: Outer Measure Interval-Length. For every interval , its outer-measure is its length, .

Outer Measuring Intervals Optional Exercises

Theorem: Outer-measure Subadditivity. Let be any countable sequence of subsets of real numbers. Then

Theorem: Null Adding and Subtracting. Let be two subsets andEa null set. Then

Outer Measure Subadditivity Optional Exercises

- Result not proved in this lesson.

Theorem: Measurable If and Only If Split. Let be a subset of real numbers. ThenEis measurable if and only if for every the setEsplits the setAcleanly.

- Results proved in this lesson.

Theorem: Null Sets Are Measurable. Every null set is measurable.

Theorem: Measurable Sets Closed Under Complement. If is a measurable subset then is measurable.

Theorem: Open Rays Are Measurable. For every the open ray is measurable.

Length Measure Optional Exercises

- Results proved in this lesson.

Theorem: The Measurable Sets Form a -algebra. The collection is a -algebra.

Theorem: Intervals Are Measurable. If is an interval then .

Sigma-algebra Optional Exercises

Theorem: Length-measure Is Countably Additive. Let be a disjoint sequence of measurable sets. Then

Theorem: Length-measure Excision. Let be two measurable sets, and . Then .

Theorem: Upward Continuity of Measure. Let be an ascending sequence of measurable sets. Then

Theorem: Downward Continuity of Measure. Let be a descending sequence of measurable sets. If is finite then

Countable Additivity Optional Exercises

### Section 2: Counter-Examples

[edit | edit source]### Section 3: Measurable Functions and Integration

[edit | edit source]### Section 4: Approximation Theorems

[edit | edit source]### Section 5: Length-integrals and their Properties

[edit | edit source]### Section 6: Derivatives of Integrals

[edit | edit source]### Section 7: Integrals of Derivatives

[edit | edit source]### Section 8: *L*^{2} Spaces

[edit | edit source]### Section 9: *L*^{p} Spaces

[edit | edit source]## Beyond Lebesgue

[edit | edit source]The course is titled "Measure Theory" and yet it discusses only Lebesgue measure. Why?

I'm trying to keep my goals modest for now -- I am writing this in my free time. I'll be lucky to complete a significant portion of this within a few months, and then I may not have time to keep up with this.

But if I find more time in the next year then I may extend this course to general measure theory.