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• Week 0 -- "Due" Thursday, September 6. I'd like you to:
  1. look up the definition of Riemann Integral in your calculus textbook and in your ERA textbook (' sold 'em for summer tan dough? check out the library or internet!!).
  2. Bring two versions of the Fundamental Theorem of Calculus to class (best found in the ERA book).
  3. Check out a bit of dirt on Riemann and Lebesgue at http://turnbull.dcs.st-and.ac.uk/history/ and save the site as a bookmark.

• Week 1 -- "Due" Thursday, September 20
  • Work Sheet 1

    For each of the following write a paragraph explaining your method of solution and in what sense it solves the problem. Of course, give your solution too!!

    1. A dart is thrown randomly at a dart board ( English Competition Board) What is the probability of hitting the red . Note, you can print the picture of the board and measure the ring sizes from the picture. Since you want probability, actual measurements are not important, only relative measurements -- so you can use the picture!
    2. The problem is that nobody wants a nuclear dump in their backyard. But Minnesota needs one, so to be fair, the state legislature (in their collective wisdom) has decided to randomly choose a location somewhere in Minnesota and build the dump there. However, they also instruct the mathematicians at St. Olaf to "determine the probability that the site chosen will be within 100 miles of a major metropolitan area." Estimate this probability.

  • Exercises
    1. State and prove both versions of the FTC.
    2. Prove that every differentiable function is continuous.
    3. Find a derivative which is not continuous. (Inherent here is that you not only give the function and find its derivative, but prove the derivative is discontinuous at some point.)
    4. Find a derivative which is not continuous at 100 points.
    5. Find a derivative which is not continuous at infinitely many points.
    6. Let C₁₀={.d₁d₂d₃... : d_i= 0 or 9 for each i}. Determine the "geometry" of C₁₀. ie locate portions of R which contain points from C and portions devoid of points of C.
    7. Find and record a proof that every continuous function defined on a compact (closed and bounded) interval is Riemann integrable.
    8. Complete the details of the arguments of Example 1.1 on page 8.
    9. Complete the details of the arguments of the two counter examples given in the third point on pages 10 and 11. e.g. for Example 1, what is the integral of each function f_n, what IS the limit function, f, and why is f not Riemann integrable?

  • Theorems and Definitions
    1. Definitions
       • null set
       • outermeasure of a set
       • monotone set function
       • countably subadditive set function
    2. Theorem Every countable set is null.
    3. Theorem If N_i is null for i=1,2,..., then U_i=1^infinity N_i is null.
    4. Theorem Outer measure is both monotone and countably subadditive.
    5. TheoremThe outermeasure of an interval is its length.

  • Week 2 Worksheet -- "Due" Thursday, September 27
    1. Read sections 2.1, 2.2 and the first three pages of 2.3
    2. Write out the proof that any countable set has measure 0.
    3. The Cantor Set, C₁₀ can be constructed using Mathematical Induction. The first step of this inductive construction is: "Delete the center 8/10 of [0,1], leaving two disjoint congruent intervals each of length 1/10."

       You are to write careful (and accurate!!) directions for the inductive step of this construction. That is, first make an appropriate Inductive Assumption. Then show how to carry out the next step using this Inductive Assumption .

    4. Prove that the Cantor Set, C₁₀, has measure zero.
    5. Let C₃={x in [0,1]: x has only 0's and 2's in its base 3 decimal expansion}. Write directions for inductively constructing C₃ and then show C₃ is a null set.
    6. Learn the proof that a countable union of null sets is null.
    7. Suppose that for each n, L_n is a length where 0< L_n<1 and that Σ L_n = M<1. Now, use an inductive construction similar to that of the one you did for C₃ and C₁₀ to construct a "Cantor Set" of measure 1-M.
    8. Write (and understand!) a complete proof that the outer measure of a closed interval is its length. (Theorem 2.3)
    9. Prove that if two sets differ by a null set, then they have the same outer measure. (This is Exercise 2.4 on page 25 in the book)
    10. Determine the smallest σ algebra, BiggyPairs, which contains all pairs of rationals. That is, every set of the form {r₁, r₂} (where r₁and r₂ are rationals) is in this σ algebra and BiggyPairs has nothing extaneous.

  • Week 3 Worksheet -- "Due" Thursday, October 4
    1. Read sections 2.3, 2.4 and 2.5
    2. Learn the proofs that null sets are measurable and that intervals are measurable.
    3. Show that if the symmetric difference of two sets is null, then the sets have the same outermeasure. That is, if m^*(AΔB)=0, then m^*(A)=m^*(B). This is Exercise 2.5 on page 25 of the book).
    4. Is the set of null sets a σ algebra? If so prove it. If not, describe how to construct the smallest σ algebra which contains the null sets.
    5. Prove that every open set can be written as a disjoint union of open intervals. Hint -- If U is an open set, define an equivalence relation on U so that the equivalence classes are open intervals.
    6. Carefully write an algorithm for how to measure any closed set. These directions should be so clearly written that your Uncle Harold could measure closed sets while watching a Viking-Packer game

  • Week 4 Worksheet -- "Due" Thursday, October 11
    1. Read sections 2.5, 2.6 and 3.1
    2. Find a derivative that is discontinuous on a closed set of positive measure. (This is a Proastie-Toastie problem! so let me know as soon as you get it to receive glory, honors as well at the PT
    3. Prove that any F_σ set, E, that does not contain an interval can be written as a disjoint union of closed sets.

       OR equivalently, that any F_σ set, E, can be written as an open set union a disjoint union of closed sets.

    4. Prove that any G_δ set which is dense in [0,1] is uncountable. Hint -- Show that such a set must contain a Cantor set.
    5. Find a G_δ that is not an open set and find an F_σ which is not a closed set.
    6. Find a G_δσ that is not a G_δ and show this is equivalent to finding an F_σδ that is not F_σ.
    7. Can you continue this "finding" one stage futher?
    8. Write a paragraph describing an algorithm for measuring an arbitrary F_σ set. As an addendum comment on how to measure any G_δ.
    9. Write a mighty clean proof that a set, E, is measurable iff for every ε>0 there is an open set U and a closed set F with "F contained in E contained in U" such that m(U\F)<ε.
    10. Write an exceptionally elegant proof that that a set, E, is measurable iff there is a G_δ set A and an F_σ set B with "B contained in E contained in A" such that m(A\B)=0.

  • Week 5 Worksheet -- "Due" Thursday, October 25
    1. Read Sections 3.2, 3.3
    2. Prove that every open set is an F_σ and conclude that every closed set is a G_δ and every G_δ is an F_σδ etc.
    3. Prove that a function is continuous iff f^-1 (closed) is closed.
    4. Prove that f is measurable iff f^-1(every open interval) is measurable.
    5. Learn the proof that if f and g are measurable then so is f+g and fg. (Extra kudos given for proving, then using the "Elegant Lemma")
    6. Find and prove a characterization of Baire 1 functions in terms of inverse images of closed sets.
    7. Prove that if f is Baire 1, then f is continuous on a dense set of points.
    8. Find a Baire 1 function that is discontinuous at a dense set of points.
    9. Prove that an arbitrary function is continuous on a G_δ set (possibly empty).

  • Week 6 Worksheet -- "Due" Thursday, November 1
    1. Read Sections 4.1-4.3
    2. Learn this statement:

       Theorem If F:RxR->R is continuous and both f and g are measurable, then F(f,g) is also a measurable function.

       Use this theorem to prove that if f and g are measurable, the, |f|, f²+g², sin(f), fg, 3f³-4f²g-7fg⁴-g⁵ are also measurable.

    3. Suppose f_n is measurable for n=1,2,...,N. Prove (Hey!! induction might be a good idea!)
       1. M(x)=max(f_n(x)) is measurable.
       2. m(x)=min(f_n(x)) is measurable.
    4. Let {f_n} be a sequence of measurable functions. Prove that each of the following functions, derived from the orgininal sequence, is also measurable. (You can use the facts above, if f and g are measurable, then so is fg,f+g, and that the pointwise limit of measurable functions is measurable.
       1. F₁(x)=sup_n{f_n(x)}
       2. F₂(x)=inf_n{f_n(x)}
       3. F₃(x)=limsup_n{f_n(x)}
       4. F₄(x)=liminf_n{f_n(x)}
    5. Let F be a closed set in [0,1], and let P_n be the regular, 1/n partition of [0,1]. Define i_n to be the number of intervals from P_n which intersect F. Prove that m(F)=limit_n( i_n/n ). Is this result true for open sets? F_σ sets??
    6. Let F be a Cantor set (nowhere dense and closed) of measure 3/4 in [0,1]. Prove that 1_F is NOT Riemann integrable on [0,1] by computing the upper Riemann integral and lower Riemann integral and showing they are different. Then show 1_F is Lebesgue integrable on [0,1] and find its integral.
    7. Let F be a Cantor set in [0,1] with contiguous intervals (open sets comprising the complement) I₁, I₂, I₃, ... . Suppose f is constant on F and f is continuous on every interval I_n. If f is bounded, prove f is integrable and write an algorithm for finding its integral.

  • Week 7 Worksheet -- "Due" Thursday, November 8
    1. Read Sections 4.4-4.6
    2. Prove that if f is bounded below (above) there is an increasing (decreasing) sequence of simple functions, {f_n} which converges pointwise to f.
    3. Use the books definition of integral to prove that integration over a set A (with finite measure) is a linear operator.
    4. Learn the statement and proof of the Lebesgue Bounded Convergence Theorem.
    5. Use the Bounded Convergence Theorem to prove the algorithm you described in Week 6, #7 actually works.

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