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--- math105-s22:s:hexokinase:start [2022/02/10 00:13]
pzhou old revision restored (2022/02/07 10:57)
+++ math105-s22:s:hexokinase:start [2026/02/21 14:41] (current)
@@ Line 150: / Line 150: @@
 Can we characterize/classify the Borel sets?
+==== Feb 18 ====
+=== Question 0 ===
+{{ :math105-s22:s:hexokinase:4.0.pdf |}}
+==== Feb 21 ====
+=== Conjectures on products ===
+Let $E \subseteq \R^m\times\R^n$.\\
+I conjecture that $E$ is measurable if and only if $E_x \subseteq \R^n$ is measurable for a.e. $x\in\R^m$.\\
+Furthermore, supposing $E$ is measurable and $\Omega \subseteq\R^m$ is the full-measure set on which $E_x$ is measurable, I conjecture that $x \mapsto m_n(E_x)$ is a measurable function and
+$$ \int_{\R^m} (x \mapsto m_n(E_x))  =  m_{m+n}(E) $$
+I think the second conjecture has a hint of Fubini.
+==== Mar 20 ====
+=== Littlewood's three principles ===
+== Principle 1 ==
+This made me wonder what exactly "regularity" means, so I looked at Wikipedia's definition, which I now provide. Given a measure space $(X, \Sigma, \mu)$ and a topological space $(X, \tau)$ (with the same underlying set), a set $E\in\Sigma$ is inner regular if
+$$ \mu(E) = \sup\{ \mu(K) \vert K\in\Sigma \textrm{ compact } \} $$
+and outer regular if
+$$ \mu(E) = \inf\{ \mu(U) \vert U\in\Sigma \textrm{ open } \} $$
+It is regular if both of these hold,
+and $\mu$ is regular if every $E\in\Sigma$ is regular.
+== Principle 2 ==
+I was initially confused by this one; using the preimage definition of continuity, I believed that $\xi_\Q$ was a counterexample since $\Q^c$ has empty interior. However, when we restrict our domain, the meaning of "open" in our domain changes, which accounts for this.
+== Principle 3 ==
+Not much to say on this one, since it came up already in HW 5.
 ===== Homework =====
 {{ :math105-s22:s:hexokinase:1.pdf |}} \\
-{{ :math105-s22:s:hexokinase:2.pdf |}}
+{{ :math105-s22:s:hexokinase:2.pdf |}} \\
+{{ :math105-s22:s:hexokinase:3.pdf |}} (slightly updated since Gradescope feedback) \\
+{{ :math105-s22:s:hexokinase:4.pdf |}} (slightly updated since Gradescope feedback) \\
+{{ :math105-s22:s:hexokinase:5.pdf |}} \\
+{{ :math105-s22:s:hexokinase:6.pdf |}} \\
+{{ :math105-s22:s:hexokinase:7.pdf |}} \\
+{{ :math105-s22:s:hexokinase:8.pdf |}} \\
+{{ :math105-s22:s:hexokinase:9.pdf |}} \\
+{{ :math105-s22:s:hexokinase:10.pdf |}}

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