math105-s22:s:mheaney:start
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| My other interests are music and sport. I play piano and saxophone, I teach piano at home and currently trying to teach myself the trumpet. I also ran the half marathon across the golden gate last November! | My other interests are music and sport. I play piano and saxophone, I teach piano at home and currently trying to teach myself the trumpet. I also ran the half marathon across the golden gate last November! | ||
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| + | **Study Methods** | ||
| + | **// | ||
| + | Lebesgue Outer Measure (Pugh Chp6 and Tao Chp 7) // | ||
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| + | **//Tool Box//** As I go through revising for exams and for problem sets. I try to put together a toolbox. Just definitions and theorems we have covered without the proofs. I'll add that here when it is complete | ||
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| 3. Take limit of these simple functions, when more points are added in the range of original functions. | 3. Take limit of these simple functions, when more points are added in the range of original functions. | ||
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| + | //**HW6 Summary of results for Lebesgue Measure**// | ||
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| + | **Outer Measure** | ||
| + | 1. From Pugh's approach, he defines the outer measure of a set using Intervals, Rectangles, and boxes. | ||
| + | Lebesgue outer measure of a set $A \subset \R$ is\\ | ||
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| + | $m^{*}A$ = $inf${ $\Sigma_{k}$ $\vert$ $I_{k}$ | : {$I_{k}$ is a covering of A by open intervals} \\ | ||
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| + | The important theorem for outer measure is proving its properties: \\ | ||
| + | //a) The outer measure of the empty set is 0, $m^{*}\emptyset$ = 0// \\ | ||
| + | //b) If A $\subset$ B then $m^{*}A$ $\leq$ $m^{*}B$// \\ | ||
| + | //c) if A = $\cup$ $A_{n}$ then $m^{*}A$ $\leq$ $\Sigma$ $m^{*}A_{n}$ \\ | ||
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| + | Another definition we continued to use throughout Lebesgue Theory was ** If $Z \subset \R^{n}$ has outer measure zero then it is a zero set** \\ | ||
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| + | 2. The second topic we learnt about was **Measurability** \\ | ||
| + | First defining **(Lebesgue) measurable** | ||
| + | A set $E \subset \R$ is Lebesgue measurable if the division $E|E^{c}$ of $\R$ is so " | ||
| + | $m^{*}X$ = $m^{*}$( X $\cap$ E) + $m^{*}$(X $\cap$ $E^{c}$) \\ | ||
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| + | The fact an empty set is measurable and that if a set is measurable then so is its compliment was needed for futher chapters as we will see... | ||
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| + | In this section we were introduced to $\sigma$ - algebra, which is a collection of sets that includes the empty set, is closed under complement and is closed under countable union. \\ | ||
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| + | 3. **Mesomorphism** \\ | ||
| + | // measure space, differences between mesemorphism, | ||
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| + | **4. Regularity** \\ | ||
| + | **Theorem 11** // Lebesgue measure is **regular** in the sense that each measurable set E can be sandwiched between an $F_{\sigma}-set$ and a $G_{\delta}-set$ , F $\subset$ E $\subset$ G , such that G\F is a zero set. Conversely, if there is such an F $\subset$ E $\subset$ G, E is measurable. // | ||
| + | Affine motions.. | ||
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| + | **__Final Essay- Fast Fourier Transforms (FFT)__** | ||
| + | I took a perspective based on my own background and how I visualize and use FT and FFT's in general | ||
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math105-s22/s/mheaney/start.1645233936.txt.gz · Last modified: 2026/02/21 14:43 (external edit)
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