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--- math105-s22:notes:lecture-1 [2022/01/17 18:33]
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+++ math105-s22:notes:lecture-1 [2022/01/19 10:15] (current)
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 ====== Lecture 1 ======
+{{ :math105-s22:note_jan_18_2022_math105_1.pdf | note}}, [[https://berkeley.zoom.us/rec/share/qx3MKU2zdiac0LzeZn5IoQZTrSgUg01JZTxsGfU7Q7oIRWCUJJRtJz4RjINQnFuA.Ty9rPvaUgOriN4WM | video]]
+  * Motivation for Lebesgue measure (read Tao 7.1)
+  * What is the definition of outer-measure?
+  * Hmm, the outer-measure of a closed box? Why is it so complicated?
+    * Interlude, the [[https://courses.wikinana.org/math105-s22/lecture-1 | Banach-Tarski sphere]].
+Discussion Time:
+  * How to prove that  $\{1,2,3\}$  has zero outer-measure?
+  * How to prove that  $\Z$  has zero outer-measure? And  $\Q$ ?
+  * Can you summarize some rules that allow you get to the above results quicker?
+  * Can you show that  $\R$  has zero outer-measure in  $\R^2$ ? Can you prove that in general, a lower dimensional 'manifold' in  $\R^n$  has zero outer measure? Like the unit circle in  $\R^2$ ?
+  * (hard) How does outer-measure behave under product? Does  $m^*(A \times B) = m^*(A) \times m^*(B)$ ?
+===== Sketch =====
 Welcome to this class. So, I assumed you all had taken math 104 or the equivalent of it, which covers sequence and limits, metric space topology (open sets, distance functions, compact sets etc), and also some Riemann integrals. Why do we want to take the second course in analysis?

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