| Date | Tentative class plan: |
| Fri, Aug 29 | Welcome! Start Lang II.1: Open and closed sets, and continuity |
| Wed, Sep 3 |
Continue Lang II.1: Open and closed sets, and continuity |
| Fri, Sep 5 |
Finish Lang II.1: Open and closed sets, and continuity Start Lang II.2: Connectedness |
| Wed, Sep 10 |
Finish Lang II.2: Connectedness Start Lang II.3: Compactness |
| Fri, Sep 12 |
Continued discussion of Lang II.3: Compactness.
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| Wed, Sep 17 |
Finished Lang II.3: Compactness.
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| Fri, Sep 19 |
Lang II.4: Separation by continuous functions.
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| Tue, Sep 23 | Friday Schedule Lang III.1: The Stone-Weierstrass Theorem. |
| Wed, Sep 24 | No classes. |
| Fri, Sep 26 | No classes. |
| Wed, Oct 1 | Metric Completions |
| Fri, Oct 3 | No classes. |
| Wed, Oct 8 |
The Baire Category Theorem and applications.
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| Fri, Oct 10 |
Lang III.3: Ascoli's Theorem.
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| Wed, Oct 15 |
The implicit function theorem.
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| Fri, Oct 17 |
The general implicit function theorem and the inverse function theorem.
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| Wed, Oct 22 |
Review homework 7. More discussion of the inverse function theorem.
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| Fri, Oct 24 | Midterm Exam |
| Wed, Oct 29 |
Pugh 6.1-6.2: The Lebesgue outer measure and other outer measures..
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| Fri, Oct 31 |
Pugh 6.1-6.2: Measures from outer measures, Properties of Lebesgue measure on R.
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| Wed, Nov 5 |
Finish up properties of Lebesgue measure including regularity (Pugh 6.3) Folland 1.4: Review of constructing measures via extension. |
| Fri, Nov 7 |
Folland 1.5: Borel measures on R.
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| Wed, Nov 12 |
Pugh 6.A: Non-measurable sets.
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| Fri, Nov 14 |
Pugh 6.4-6.5: The Lebesgue integral.
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| Wed, Nov 19 |
Equivalence of notions of measurability.
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| Fri, Nov 21 |
Properties of integrals (e.g., Monotone convergence theorem, Dominated convergence theorem)
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| Wed, Nov 26 |
Folland 0.4: The ordinals The Borel Hierarchy |
| Fri, Nov 28 | Thanksgiving Recess. |
| Wed, Dec 3 |
Pugh 6.6: Cavalieri's principle and Fubini's theorem. Pugh 6.7: Vitali coverings and density points. |
| Fri, Dec 5 |
L1, step (or simple) functions, and Lusin's theorem
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| Wed, Dec 10 |
Density of continuous functions in L1. Problems. |
| Fri, Dec 12 |
Comparison to Riemann integral, relation between differentiation and integration, bounded variation and absolute continuity.
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