New MT Article Ideas: Difference between revisions

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==Measurable Sets==
==Measurable Sets==
* Vitali's Theorem and non-existence of a measure on <math>2^\mathbb{R}</math>
* Borel <math>\sigma</math>-algebra and <math>\mathcal{B}_\mathbb{R}</math>
* Product <math> \sigma </math> - algebra
* Product <math> \sigma </math> - algebra
* Borel measureable sets vs. Lebesgue measurable sets
* Borel measureable sets vs. Lebesgue measurable sets


==Measures==
==Measures==
* Caratheodory's Theorem
* Isomorphism of Measure Spaces
* Lebesgue-Stieljes Measures
* Characterization of Finite Borel Measures
* Characterization of Finite Borel Measures
* Approximation by Open/Compact Sets
* Result connecting the absolute continuity of a Borel measure with respect to the Lebesgue measure with absolutely continuous functions
 
==Measurable Functions==
* Simple functions
* Measurable functions


==Integration of Nonnegative Functions==
==Integration of Nonnegative Functions==
* Monotone Convergence Theorem
* Key counterexamples
* Key counterexamples


==Integration of Real-Valued Functions==
* Dominated Convergence Theorem
* L1 as a normed vector space


==Modes of Convergence==
==Modes of Convergence==
* Pointwise a.e. convergence
* Pointwise a.e. convergence
* L1 convergence
* L1 convergence
* Convergence in measure
* Egorov's theorem
* Lusin's theorem

Latest revision as of 22:32, 19 December 2020

Here is a list of new article ideas for the Measure Theory Wiki.

Analysis Fundamentals

  • Metric space vs. Pseudometric space

Measurable Sets

  • Product - algebra
  • Borel measureable sets vs. Lebesgue measurable sets

Measures

  • Characterization of Finite Borel Measures
  • Result connecting the absolute continuity of a Borel measure with respect to the Lebesgue measure with absolutely continuous functions

Integration of Nonnegative Functions

  • Key counterexamples


Modes of Convergence

  • Pointwise a.e. convergence
  • L1 convergence