Measure Theory

Design Principles

This course is designed with a few principles in mind.

Natural Proofs

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

• 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

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.

• Lexicon

• Section 1 Proofs, Measure

• Section 2 Proofs, Counter-examples in Length Measurement

• Section 3 Proofs, Measurable Functions and Integration

• Section 4 Proofs, Approximation Theorems

• Section 5 Proofs, Length Integrals and Their Properties

• Section 6 Proofs, Derivatives of Integrals

• Section 7 Proofs, Integrals of Derivatives

• Section 8 Proofs, L² Spaces

• Section 9 Proofs, L^p Spaces

Lesson Schedule

Section 0: Introduction

• Section 0, Lesson 0: Introduction to the Course

Introduction Optional Exercises

• Section 0, Lesson 1: Fourier and the Need for Swaparoo

Fourier Optional Exercises

Section 1: Measuring Sets of Reals

• Section 1, Lesson 0: Measuring Sets of Real Numbers

Measuring Reals Optional Exercises

• Section 1, Lesson 1: There Is No Total Measure of Real Numbers

Results proved in this lesson.

Theorem: There Is No Total Measure of Real Numbers

Let $\mu :𝒫(ℝ)\to ℝ$ be any real-valued set function.  Then $\mu$ must not satisfy at least one of the properties: Length-measure, nonnegativity, translation-invariance, countable additivity.

No Total Measure Optional Exercises

• Section 1, Lesson 2: Outer Measure

Results proved in this lesson.

Theorem: Outer-measure Is Well-defined and Nonnegative.

For every $A\subseteq ℝ$ the outer-measure takes a unique extended real number value.  That is to say ${\lambda }^{*}(A)\in {ℝ}^{*}$.  Also this value is nonnegative, $0\le {\lambda }^{*}(A)$.  

Theorem: The outer measure by countable sums.

${\lambda }^{*}(A)=\{\sum _{I\in 𝔒}\ell (I):𝔒\text{ is a countable open-interval over-approximation of }A\}$

Theorem: Outer-measure Is Translation Invariant.

For every subset $A\subseteq ℝ$ and real number $c\in ℝ$ the outer-measure of A is invariant under translation by c, 
  ${\lambda }^{*}(A)={\lambda }^{*}(A+c)$

Theorem: Outer-measure Is Monotonic.

If $A\subseteq B\subseteq ℝ$ then ${\lambda }^{*}(A)\le {\lambda }^{*}(B)$.

Theorem: Countable Sets Are Null.

If $A\subseteq ℝ$ is a countable set, then ${\lambda }^{*}(A)=0$.

Outer Measure Optional Exercises

• Section 1, Lesson 3: Outer Measuring Intervals

Results proved in this lesson.

Theorem: Outer Measure Interval-Length.

For every interval $I\subseteq ℝ$, its outer-measure is its length, ${\lambda }^{*}(I)=\ell (I)$.

Outer Measuring Intervals Optional Exercises

• Section 1, Lesson 4: Outer Measure Subadditivity

Theorem: Outer-measure Subadditivity.

Let $A_1,A_2,\dots \subseteq ℝ$ be any countable sequence of subsets of real numbers.  Then 
  ${\lambda }^{*}\left(\bigcup _{k=1}^{\mathrm{\infty }}{A}_k\right)\le \sum _{k=1}^{\mathrm{\infty }}{\lambda }^{*}(A_k)$

Theorem: Null Adding and Subtracting.

Let $A,E\subseteq ℝ$ be two subsets and E a null set.  Then 
  ${\lambda }^{*}(A)={\lambda }^{*}(A\cup E)={\lambda }^{*}(A\setminus E)$

Outer Measure Subadditivity Optional Exercises

• Section 1, Lesson 5: Length Measure

Result not proved in this lesson.

Theorem: Measurable If and Only If Split.

Let $E\subseteq ℝ$ be a subset of real numbers.  Then E is measurable if and only if for every $A\subseteq ℝ$ the set E splits the set A cleanly.  

Results proved in this lesson.

Theorem: Null Sets Are Measurable.

Every null set is measurable.

Theorem: Measurable Sets Closed Under Complement.

If $E\subseteq ℝ$ is a measurable subset then $E^c$ is measurable.

Theorem: Open Rays Are Measurable.

For every $a\in ℝ$ the open ray $(a,\mathrm{\infty })$ is measurable.  

Length Measure Optional Exercises

• Section 1, Lesson 6: The Measurable Sets Form a Sigma-algebra

Results proved in this lesson.

Theorem: The Measurable Sets Form a $\sigma$-algebra.

The collection $ℳ$ is a $\sigma$-algebra.

Theorem: Intervals Are Measurable.

If $I\subseteq ℝ$ is an interval then $I\in ℳ$.

Sigma-algebra Optional Exercises

• Section 1, Lesson 7: Properties of Length-measure

Theorem: Length-measure Is Countably Additive.

Let $E_1,E_2,\dots \in ℳ$ be a disjoint sequence of measurable sets.  Then 
  $\lambda \left(\underset{k=1}{\overset{\mathrm{\infty }}{⨆}}{E}_k\right)=\sum _{k=1}^{\mathrm{\infty }}\lambda (E_k)$

Theorem: Length-measure Excision.

Let $E,F\in ℳ$ be two measurable sets, and $E\subseteq F$.  Then $\lambda (F\setminus E)=\lambda (F)-\lambda (E)$.

Theorem: Upward Continuity of Measure.

Let $E_1,E_2,\dots \in ℳ$ be an ascending sequence of measurable sets.  Then 

  $\underset{n\to \mathrm{\infty }}{\lim }\lambda (E_m)=\lambda \left(\bigcup _{k=1}^{\mathrm{\infty }}{E}_k\right)$

Theorem: Downward Continuity of Measure.

Let $E_1,E_2,\dots \in ℳ$ be a descending sequence of measurable sets.  If $\lambda (E_1)$ is finite then 
  $\underset{n\to \mathrm{\infty }}{\lim }\lambda (E_m)=\lambda \left(\bigcap _{k=1}^{\mathrm{\infty }}{E}_k\right)$

Countable Additivity Optional Exercises

Section 2: Counter-Examples

• Section 2, Lesson 0: Counter-examples

Section 3: Measurable Functions and Integration

• Section 3, Lesson 0: Length-integration and Measurable Functions

• Section 3, Lesson 1: Length-integration Defined

Section 4: Approximation Theorems

• Section 4, Lesson 0: Integration Is Hard

• Section 4, Lesson 1: Approximations of Measurable Sets

• Section 4, Lesson 2: Approximations of Measurable Functions

• Section 4, Lesson 3: Approximations of Sequences of Functions

Section 5: Length-integrals and their Properties

• Section 5, Lesson 0: Length-measure Integration Is the One that We Want

• Section 5, Lesson 1: Properties of Simple Integrals

• Section 5, Lesson 2: Foundational Properties of Bounded Integrals

• Section 5, Lesson 3: Basic Properties of Bounded Integrals, and the Bounded Convergence Theorem

• Section 5, Lesson 4: Properties and Convergence of Nonnegative Integrals

• Section 5, Lesson 5: Consequences of the MCT

• Section 5, Lesson 6: Properties of General Integrals, and Lebesgue's Convergence Theorem

• Section 5, Lesson 7: Convergence in Measure

Section 6: Derivatives of Integrals

• Section 6, Lesson 0: Differentiation and Integration

• Section 6, Lesson 1: Integrable Functions Are Almost Continuous

• Section 6, Lesson 2: Markov and Hardy-Littlewood

• Section 6, Lesson 3: Derivatives of Integrals

Section 7: Integrals of Derivatives

• Section 7, Lesson 0: Integrating Derivatives

• Section 7, Lesson 1: Monotone Functions Are Differentiable A.E.

• Section 7, Lesson 2: Bounded Variation

• Section 7, Lesson 3: The Devil's Staircase

• Section 7, Lesson 4: Absolute Continuity

Section 8: L² Spaces

• Section 8, Lesson 0: L² Inner Product, Norm, Space, Distance

• Section 8, Lesson 1: L² Is a Vector Space

• Section 8, Lesson 2: L² Is an Inner Product Space

• Section 8, Lesson 3: L² Is Complete

• Section 8, Lesson 0: Quadratic Forms, Integral Equations, Expectation, Variance

Section 9: L^p Spaces

• Section 9, Lesson 0: Generalizing to L^p

• Section 9, Lesson 1: Convexity and the Product-to-Sum Bound

• Section 9, Lesson 2: Integral Product Bound and Triangle Inequality

• Section 9, Lesson 3: L^∞

Beyond Lebesgue

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.