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math105-s22:s:rasmuspallisgaard:start [2022/03/07 18:44]
pallisgaard [Homework] 		math105-s22:s:rasmuspallisgaard:start [2026/02/21 14:41] (current)
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 {{ :math105-s22:s:rasmuspallisgaard:maht105-hw5-rasmus-pallisgaard-2.pdf |Homework 5}} {{ :math105-s22:s:rasmuspallisgaard:maht105-hw5-rasmus-pallisgaard-2.pdf |Homework 5}}
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 +{{ :math105-s22:s:rasmuspallisgaard:analysis_ii_-_notes_3.pdf |Homework 6}} (There is a mistake in problem 1, b. I'll fix it and update the file.)
  
 **Resume of Lebesque measure theory** **Resume of Lebesque measure theory**
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 Proving the other direction, if $E$ has measure zero, then there exists a $G_\delta$-set $G\supset E$, The main step is to set up $X(\alpha)=\{x:m(G_x)>\alpha\}$ for each slice $G_x$. One can then finalise the proof using the same disjointizing method on a compact set $K(x)$ contained in $G_x$ with $m(K(x))=m(G_x)$ and a neighbourhood $W(x)$. Proving the other direction, if $E$ has measure zero, then there exists a $G_\delta$-set $G\supset E$, The main step is to set up $X(\alpha)=\{x:m(G_x)>\alpha\}$ for each slice $G_x$. One can then finalise the proof using the same disjointizing method on a compact set $K(x)$ contained in $G_x$ with $m(K(x))=m(G_x)$ and a neighbourhood $W(x)$.
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 +{{ :math105-s22:s:rasmuspallisgaard:math105-hw5-rasmus-pallisgaard.pdf |Homework 5}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105_hw6_rasmus_pallisgaard.pdf |Homework 6}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105-hw7-rasmus-pallisgaard.pdf |Homework 7}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105_hw8.pdf |Homework 8}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105_hw9.pdf |Homework 9}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105_hw10.pdf |Homework 10}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105_hw11.pdf |Homework 11}}
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 +{{ :math105-s22:s:rasmuspallisgaard:math105-hw12-rasmus-pallisgaard.pdf |Homework 12}}
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 +====Final Essay====
 +Here is my final essay on lebesque integration and measure theory, and why its needed and relevant in the context of integration.
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 +{{ :math105-s22:s:rasmuspallisgaard:final_essay.pdf |Why Do We Need Measure Theory?}}
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math105-s22/s/rasmuspallisgaard/start.1646678676.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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