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. - Gδ set and Fσ 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 Measures. - 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 - 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 Lebesgue 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 A.E. Dominated Convergence Theorem - 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 Theorem # 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 on h-intervals. - Then showing the Existence of the Borel 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 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 infinity 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