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math105-s22:s:hexokinase:start

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math105-s22:s:hexokinase:start [2022/02/21 13:32]
hexokinase [Feb 21] 		math105-s22:s:hexokinase:start [2026/02/21 14:41] (current)
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 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 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
 $$ \int_{\R^m} (x \mapsto m_n(E_x))  =  m_{m+n}(E) $$ $$ \int_{\R^m} (x \mapsto m_n(E_x))  =  m_{m+n}(E) $$
-I think this second conjecture has a hint of Fubini.+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 ===== ===== Homework =====
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-{{ :math105-s22:s:hexokinase:3.pdf |}} \\ +{{ :math105-s22:s:hexokinase:3.pdf |}} (slightly updated since Gradescope feedback) \\ 
-{{ :math105-s22:s:hexokinase:4.pdf |}}+{{ :math105-s22:s:hexokinase:4.pdf |}} (slightly updated since Gradescope feedback) \\ 
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 +{{ :math105-s22:s:hexokinase:9.pdf |}} \\ 
 +{{ :math105-s22:s:hexokinase:10.pdf |}}
math105-s22/s/hexokinase/start.1645450376.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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