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--- math105-s22:s:frankwang:start [2022/03/20 00:49]
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+**Homeworks**
 {{ :math105-s22:s:frankwang:105hw1.pdf |}}
+**Final Essay on some counterexamples in measure theory:**
+{{ :math105-s22:s:frankwang:some_counterexamples_.pdf |}}
 **Summary of Lebesgue Integral**
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 **Summary of Littlewood's Three Principles**
-Littlewood's First Principle states
+Littlewood's First Principle states that given $\epsilon > 0$, and some measurable $E \subset [a, b]$ contains a compact subset covered by finitely many intervals, and the union of these intervals differs from E by a set of measure less than $\epsilon$. Another way to state this is that the set $E$ is a finite union of intervals, except for an $\epsilon$-set.
+Littlewood's Second Principle states that if you have a measurale function, then for any $\epsilon > 0$, removing a set of measure $\epsilon$ will result in a continuous function.
+Littlewood's Third Principle states that if you have a sequence of measurable functions mapping an interval $[a, b]$ to $\mathbb{R}$, which converges almost everywhere, then except for a set of measure $\epsilon$, the sequence converges uniformly.
+Littlewood's Second Principle is called Lusin's Theorem and Littlewood's Third Principle is called Egoroff's Theorem.

Lecture Notes