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Measure and Integration Paper Information:
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Select a topic for a paper and partners to work on the topic with you. Although I would prefer you work in groups of 3, I am open to having you work by yourself or in a group of 2 or even 4. See me, though before selecting a non-3 person option. Each paper should be written in a clear mathematical style; you can use our book as a template format. You'll want to include introductory material, historical connections, theorem statements, proofs, and transitional material in each section. In a 10 page paper you'll want to have:

- An Introduction (~1.58 pages)
- 2 or 3 Main Sections (~2.34-3.06 pages each)
- A Concluding Section or Wrapup Section (~.82 pages)
- A Reference Section. Click Here for formatting examples.

- Turn in your paper
- Give a ~20 minute talk to the rest of the class

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Possible Measure and Integration Paper Topics:
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- B&B, Not Just Bed and Breakfast Anymore
Develop the Borel sets and show they're measurable. Give examples to show that F

_{σ}is not a subset of G_{δ}and vica versa. Give algorithms for measuring open, closed, F_{σ}, and G_{δ}sets. Define the heierarchy of Baire functions and prove that f is in B_{1}if and only if f^{-1}(open) is an F_{σ}set. Prove that every Baire function is measurable and that every derivative is B_{1}. Prove that every bounded Baire function is integrable and as a corollary that every bounded derivative is integrable and that the integral is the original function. Show that each Baire Class is a vector space. - Daring to be a Derivative
There are derivatives that are not integrable in either the Riemann sense or in the Lebesgue sense. But, every bounded derivative is Lebesgue integrable. First, describe a bounded derivative that is not Riemann integrable. Then locate a derivative that is not Lebesgue integrable. This is so non-intuitive that careful proofs are required

- Not in Riemann Land Anymore
Here you'll want to compare and contrast the Riemann and Lebesgue integrals finding the exact condition a function must have to be Riemann integrable. You'll also want to find functions which are integrable in the Lebesgue sense but not in the Riemann sense.

- Convergence and More Convergence
Carefully develop the notions of pointwise and uniform convergence of sequences of functions. Give examples of sequences which converge pointwise but not uniformly. State and prove the theorems of

**Egoroff**and**Lusin**; give examples to show how they work. Find a sequence of Riemann integrable functions which converges pointwise to a bounded function which is not Riemann integrable. State and prove the Lebesgue Bounded Convergence Theorem and use it to prove that if a bounded sequence of Riemann integrable functions converges poitwise to a function, f, then f is Lebesgue integrable and int(f)=limit int(f_{n}). Prove the**Lebesgue Dominated Convergence Theorem**using Egoroff's Theorem. - Apples and Oranges?
Compare and contrast the Riemann and Lebesgue integrals. What are THE properties that makes them "integrals". What other properties do they share and how do they differ. What functions or classes of functions are Lebesgue integrable but not Riemann integrable? Characterize the functions which are Riemann integrable and prove they are a vector space. Prove that any Riemann integrable function is Lebesgue integrable and the integrals are the same. State and prove the strongest versions of the FTC you can find for each of these two integrals.

- Pardon my Probability
First define "probability space" and develop the notions of "probability measure" (or "probability distribution") and "random (real) variable"; develop some examples. Discuss the notions of "joint probability distribution", and "independent" and "dependent" "random variables"; illustrate with some examples. Define and explore the "Rademacher Functions" explaining what they are and why they are important. Define "standard deviation", "variance" and "expectation." What does it mean for random variables to be "identically distributed?" What is a "distributional measure" and what are some examples of distributional measures?

- Living Large
First define "probability space" and develop the notions of "probability measure" (or "probability distribution") and "random (real) variable"; develop some examples. Discuss the notions of "joint probability distribution", and "independent" and "dependent" "random variables"; illustrate with some examples. Now state and prove the

*Borel-Cantelli Lemmas*and investigate the*Weak*and*Strong Laws of Large Numbers.* - From the Ground Up
First give a full historical account of Lesbesgue's motivation for developing the integral named after him. Make certain to give historical context for his work and point to other significant (non-mathematical) events that occurred during the time he was doing this work. Then, carefully develop the Lebesgue integral and it's basic properties. Show how Lebesgue measure is integral to the definition of integral (sorry!!). Make sure you show how to integrate 1. bounded measurable functions 2. measurable functions which are bounded below 3. unbounded measurable functions.

- Norm and Trig: A Love Story
Develop the basic definitions and properties of L^2[-π,π]. Prove that it's a normed vector space, derive formulas for the distance between functions and the angle between functions. Show that the Trig functions are an orthonormal set in L^2[-π,π]. Explain how to find the Fourier coordinates (coefficients) of a given function f. Compute an example which shows that the function need not coincide with its Fourier Series. Prove that if f is in L^2[-π,π] then the Fourier Series for f converges to f a.e.

- Integral Into Space; Infinity and Beyond!
Although the Lebesgue integral does integrate every bounded derivative, it does not integrate

**every**derivative. But the Henstock-Kurzweil integral does exactly that and more. In this topic you'll develop the basic notions of the H-K integral and see how it manages to integrate unbounded functions Lebesgue cannot. You'll also see why it's the same as the Lebesgue integral when that integral does exist. - Those Cantankerous Non-Measurable Sets
There is a plethora of non-measurability out there. here you'll find several methods of constructiong non-measurable sets and learn a bit of set theory to boot.

- Into the Center: A Journey to the Central Limit Theorem
The CLT is actually not "A" theorem, but a category of theorems about attractor distributions for iid random variables. This topic is not for the faint hearted but understanding what the CLT has to say is, well ... pretty central to understanding the main applications of probability theory.

- Exalted Esoteric Ergotic Theory
Ergodic Theory was a surprise and the entire study is a consequence of an intuitively clear result proved wrong! Define

*Measure Space*and*Measure Preserving Transformations*. What is a "time" average and what is a "spacial average"? How does one compute these and what are they called what they are? Decipher and discuss the*Ergodic Theorem(s)*. Why is this theorem useful and why was it controversial? Give plenty of cool examples and explore each carefully.