Week 1

Theme

The main theme of this portion of the course is to make sense of the integral $$\int\limits _{X}f \, d\mu$$more precisely we want to define the Lebesgue Integral where $f:X\to \mathbb{C}$ is a Measurable Function and $\mu$ is a Measure, $X$ is a Metric Space and overall our motivation is to address the shortcomings of the Riemann Integral.

Content covered

• We reviewed the Riemann Integral:
  • We defined the Darboux sums, the Mesh and used these to determine what it means to be Riemann integrable.
• We introduced the idea of the Lebesgue Integral
  • We noted that we needed additional tools to formally define it though
• Enter Measure Theory
  • We first introduced Topological Spaces and their constituent parts
    • What is a Topology? What is an Open set?
    • What is an Induced Topology, what are examples of different topologies (e.g. Standard topology)?
    • This theory is important to formally define continuity and Convergence
  • We then introduced Metric Spaces and Metrics
    • We showed that every metric space has a topology, the Metric Topology.
  • We defined Convergence and continuity.
  • Defined Algebras, σ-algebras, Measurable Spaces, Measurable Functions

Week 2

Theme

Here we looked at proving a lot of propositions and results to do with measurability and continuity.

Content covered

• We first defined many measure theory results:
  • We showed that the Composition of Measurable Functions is measurable. We then used this to show Theorem 1.8 to then show Closure of Measurability under different functions.
  • We proved a Criterion for Measurability allowing us to classify Measurable Functions easier.
  • We defined what a σ-algebra of subsets of was to help in the proof of Closure of Measurability.
  • We then defined what a Smallest σ-algebra was and also proved the Existence of smallest σ-algebra.
• We then moved into Borel theory:
  • We defined the Borel σ-algebra which allowed us to circumvent Measurable Spaces and work directly with Topological Spaces.
  • $G_\delta$ set and $F_\sigma$ set
  • Then we defined Borel function
  • We then proved some results for Measurability Criterion for Topological Codomain
  • We also looked at Limit Superior and Limit Inferior as well as their properties and then proved a result on Supremum & Infimum Preserve Measurability as well as Limit of Measurable Functions is Measurable

Week 3

Theme

Here we got more in the weeds on measures and simple functions as this allowed us to now get into defining preliminary versions of our Lebesgue Integral.

Content Covered

• We began by introducing simple functions and measures:
  • Introduced Simple Functions and showed that sequences of Simple functions converge to measurable functions.
  • We then defined what a Measure is and showed some Properties of Measure.
  • We then defined the Dirac Measure, Atomic Measure, Counting measure, and the Law.
  • We then defined the Lebesgue Measure and the Lebesgue-Stieltjes Measure.
• We then entered into defining integration
  • First we defined Integral of Simple Functions, then showed Monotonicity for integrals of simple functions
  • Second, we defined the Integral of Positive Measurable Functions and showed some Properties of Lebesgue Integral.

Week 4

Theme

We continued our discussion on the integral, proving all the convergence theorems.

Content Covered

• We began with showing some critical results in integration theory
  • We began with showing the Lebesgue Integral is a measure (Simple)
  • Then proved the Monotone Convergence Theorem
  • Then we showed the Beppo Levi Theorem
  • Then we showed Fatou’s Lemma
  • Then we showed the Lebesgue Integral is a measure (measurable)
• We then moved on to complex valued integration
  • We defined Integrable functions and the space they lie within, $\mathscr{L}^{1}(X,\mathscr{M},\mu)$.
  • Then we defined the Complex Lebesgue Integral. and the Integral Triangle Inequality
  • Finally we proved the Dominated Convergence Theorem

Week 5

Theme

This week was all about sets of measure zero

Content Covered

• We defined the idea of Almost Everywhere, then stated the Almost Everywhere DCT
• We then started conversation on sets of measure zero

  • We first defined a Complete Measure Space and showed that Every measure space has a completion

  • We then proved A.E. Beppo Levi

    Week 6

    Theme

    We capped off our theory of integration in style. We then began our process of moving on to constructing measures on a σ-algebra of subsets of $X$ and proving Hopf’s Extension Theorem

    Content Covered

• Showed the result on Integrals of Functions that are Zero a.e.
• Then we began our theory on constructing measures

  • We defined an Outer Measure on $X$.

  • Then we define what a Sequential Covering Class was

  • Using the above two definitions we constructed the Lebesgue Outer Measure and defined $\mu^{*}$-measurable

  • This allowed us to then get to the core of our theorems, beginning first with Carathéodory Theorem, then defined the Pre-measure, then some Construction of Outer Measure from Pre-measure which ultimately allowed us to state and prove Hopf’s Extension Theorem.

    Week 7

    Theme

    Let $\mathcal{B}$ be the Borel σ-algebra of $\mathbb{R}$, and let $\mu:\mathcal{B}\to[0,+\infty)$ be a finite Measure. Consider the distribution function of $\mu$, that is $F:\mathbb{R}\to[0,+\infty):F(x)=\mu((-\infty,x])$. Then

1. $F(\cdot)$ is non-decreasing
2. $F(\cdot)$ isRight Continuous
3. For $a<b$: $\mu((a,b])=F(b)-F(a)$ We want to turn this process around: given $F:\mathbb{R}\to \mathbb{R}$ increasing and Right Continuous, we want to build a Measure on $\mathcal{B}$ s.t. $\forall a<b$ we have $$\mu((a,b])=F(b)-F(a)$$Where the special case $F(x)=x$ yields the Lebesgue Measure

i.e. we want to use the results from the previous week to construct the Lebesgue-Stieltjes Measure and the Lebesgue Measure

Content Covered

• We defined h-intervals and the Collection of all finite disjoint h-intervals.
• Then showed our Distribution function is a pre-measure.
• Then showing the Existence of Lebesgue-Stieltjes Measure allowed us to:

  • Formally define the Lebesgue-Stieltjes Measure.

  • then show some results on the L-S measure such as:

  • Lebesgue-Stieltjes on Open intervals

  • Lebesgue-Stieltjes Measures are Regular

  • and Simplicity of Lebesgue-Stieltjes Measurable Sets

    Week 8

    Theme

    This week was all about convexity.

    Content Covered

• We defined Convex Function functions, then showed that Convexity implies continuity
• we then proved Jensen’s Inequality, Hölder’s Inequality, and Minkowski’s Inequality
• We capped it off with defining Lp Spaces

Week 9

Theme

We delved into Lp spaces and showed some properties

Content Covered

• We defined the Lp Norm, and then Essential bound to define the Infinity Norm.
• We defined the $L^{\infty}$ space.
• Then we proved Lp is Banach for all $p$.
• Then finally we proved S is dense in Lp and Continuous and Compactly supported functions dense in Lp
• Then we stated the Radon-Nikodym Theorem

Week 10

Theme

We capped our work on constructing measures using functions, bringing in the notion of derivatives and allowing us to create a framework that combined our Lebesgue-Stieltjes Measure, Lebesgue Measure, and define densities and distributions.

Content Covered

• Before proving Radon-Nikodym Theorem we defined what a Concentrated measure was as well as what Mutually Singular measures were.
• We then stated the RN theorem in the 2-step form.
• Then we showed that the for any absolutely continuous $F$, the Lebesgue-Stieltjes Measure $\mu_{F}$ is absolutely continuous w.r.t. $m$, the Lebesgue Measure
• This led us to show that Radon-Nikodym Derivative is the Density i.e.
  • Any non-decreasing absolutely continuous $F$ is differentiable a.e. and in this case, the Radon-Nikodym Derivative of $\mu_{F}$ w.r.t. $m$ is actually the derivative of $F$:$$\frac{d\mu_{F}}{dm}=F'$$
• To prove this we introduced Lebesgue points and sets that Shrink nicely. This allowed us to prove some intermediate results such as Almost every point is a Lebesgue point, and redefining Lebesgue points in terms of sets that shrink nicely to then show the Equivalence between Density and Radon-Nikodym Derivative (i.e. for $f$ s.t. $F=\int\limits f \, dm$, then $f=F'$).

Week 11

Theme

this week was all about Fubini and Tonelli.

Content Covered

• We capped off our Radon-Nikodym Theorem stuff
• We then started our venture into Fubini-Tonelli
  • We defined Measurable Rectangles, Elementary Sets, the Product σ-algebra $\mathscr{P}$ and the concept of a Monotone Class.
  • We then proved $\mathscr{P}$ is the smallest monotone class containing $\mathcal{G}$
  • We then proved some preliminary results like Slices of P stay in their respective σ-algebras and Slices of product measurable function are in measurable in resultant σ-algebras
  • This allowed us to prove Fubini Theorem (For indicator functions)
  • then we finally proved the Fubini-Tonelli