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math105-s22:s:frankwang:start

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math105-s22:s:frankwang:start [2022/03/06 23:28]
frankwang 		math105-s22:s:frankwang:start [2026/02/21 14:41] (current)
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 +**Homeworks**
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 {{ :math105-s22:s:frankwang:105hw1.pdf |}} {{ :math105-s22:s:frankwang:105hw1.pdf |}}
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 +**Final Essay on some counterexamples in measure theory:**
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 +{{ :math105-s22:s:frankwang:some_counterexamples_.pdf |}}
  
 **Summary of Lebesgue Integral** **Summary of Lebesgue Integral**
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 This setup allows Tao to define Lebesgue integration in Chapter 8. By introducing simple functions, functions which have finitely many values in their images, and defining the Lebesgue integral for simple functions, we can then define the Lebesgue integral for any non-negative measurable function by taking the supremum of the integral of a simple function which is dominated by the function. From there, we can integrate any absolutely integrable function, which is a function which has a finite Lebesgue integral of its absolute value. If this is the case, we define the integral to be the integral positive parts minus the integral of the absolute value of the negative parts. This setup allows Tao to define Lebesgue integration in Chapter 8. By introducing simple functions, functions which have finitely many values in their images, and defining the Lebesgue integral for simple functions, we can then define the Lebesgue integral for any non-negative measurable function by taking the supremum of the integral of a simple function which is dominated by the function. From there, we can integrate any absolutely integrable function, which is a function which has a finite Lebesgue integral of its absolute value. If this is the case, we define the integral to be the integral positive parts minus the integral of the absolute value of the negative parts.
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 +**Summary of Littlewood's Three Principles**
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 +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.
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 +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.
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 +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.
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 +Littlewood's Second Principle is called Lusin's Theorem and Littlewood's Third Principle is called Egoroff's Theorem.
  
math105-s22/s/frankwang/start.1646609335.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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