Monday September 11:
- Read: Review the Riemann integral: $\mathcal R\int_a^b
f(x)\,dx$.
- HW: Work problems 1, 2 and 3 from the
First homework
assignment. (The assignment was last modified: 11:51 am, July 15, 2022).
- If you want the source for the first homework assignment, it
is here. You'll need my personal exam
class: dpwexam-new.cls. Let me know
if there additional missing macro files.
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Update: Wednesday, September 13:
- Read: We're more or less in the very beginning of Chapter 1 of Rudin.
- Homework: Now you're ready for problems 4, 5 and 6.
- Optional: Here's an optional
little worksheet on Borel sets. Let
me know if you want to talk about it.
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Some solutions:
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Saturday, September 23:
- Homework: Here is the second
assignment. You should start on problems 1--4.
- E-Mail: I tried to use canvas to circulate this
assignment on Saturday morning. If you didn't see the email, you
need to tweak your canvas settings.
- Here is the LaTeX source for
homework #2.
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Sunday October 1:
- Homework #2 You should complete the second homework
assignment by Friday. (Really, it should be Wednesday, but I
forgot to say so earlier.)
- Not to be turned in: Let $f\in \mathcal{L}^1(X)$ and
define $\nu: \mathcal{M}\to \mathbf{C}$ by
$$\nu(E)=\int_E f(x)\,d\mu(x).$$
Show that $\nu$ is a complex measure on $(X\mathcal M)$. Show
directly (without invoking Hahn/Jordan decompositions) that there
are finite (positive) measures $\mu_i$ such that $$\nu(E)=
\mu_1(E) -\mu_2(E) +i \bigl(\mu_3(E) - \mu_4(E)\bigr)$$ with
$\mu_1\perp \mu_2$ and $\mu_3\perp \mu_4$.
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Some Date in early October:
- Homework: Here is the third
assignment. I'm a bit late with this, so all problems are
in play. I don't think that the source will be available this
time.
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Monday, October 16th:
- Homework: Here is the fourth
assignment. You should start on problems 2--8 right away.
This is technically "backgroud material", but it will be new to
some of you. It is certainly good background for the written
quals.
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October 21:
- Homework: Here are some solutions
for the third
assignment.
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Monday, October 30:
- Last Homework fifth homework.
Some of the Rudin problems are trickly. Feel free to ask
questions!
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November 1:
- Homework: Here are some solutions
for the fourth
assignment.
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Tuesday 14 November:
- FINAL EXAM final exam.
Sorry for the delay in posting!
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