Difference between revisions of "Measure Theory"
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(Created page with "==First things== * Ring of sets * Algebra of sets * Sigma-ring * Sigma-algebra Category:Measure Theory") |
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* [[Sigma-algebra]] | * [[Sigma-algebra]] | ||
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| + | ==Measures== | ||
| + | To start with we define [[Ring of sets|rings]], for example consider the ring of all half-open-half-closed rectangles of dimension {{M|n}}, call this <math>\mathcal{J}^n</math> | ||
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| + | <math>[[a,b))\in\mathcal{J}^n</math> means <math>[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\in\mathcal{J}^n</math> | ||
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| + | This is clearly a ring, but not a [[Sigma-ring|{{Sigma|ring}}]] as for example <math>\bigcup^\infty_{n=1}[[0,1-\tfrac{1}{n}))=[[0,1]]\notin\mathcal{J}^n</math> | ||
[[Category:Measure Theory]] | [[Category:Measure Theory]] | ||
Revision as of 19:08, 15 March 2015
First things
Measures
To start with we define rings, for example consider the ring of all half-open-half-closed rectangles of dimension [ilmath]n[/ilmath], call this [math]\mathcal{J}^n[/math]
[math][[a,b))\in\mathcal{J}^n[/math] means [math][a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\in\mathcal{J}^n[/math]
This is clearly a ring, but not a [ilmath]\sigma[/ilmath]-ring as for example [math]\bigcup^\infty_{n=1}[[0,1-\tfrac{1}{n}))=[[0,1]]\notin\mathcal{J}^n[/math]
