Measures (measure theory)

Outer measure

In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory. Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in or balls in . One might expect to define a generalized measuring function on that fulfills the following requirements: 1. * Any interval of reals has measure 2. * The measuring function is a non-negative extended real-valued function defined for all subsets of . 3. * Translation invariance: For any set and any real , the sets and have the same measure 4. * Countable additivity: for any sequence of pairwise disjoint subsets of It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property. (Wikipedia).

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Micrometer/diameter of daily used objects.

What was the diameter? music: https://www.bensound.com/

From playlist Fine Measurements

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Measure Theory 2.1 : Lebesgue Outer Measure

In this video, I introduce the Lebesgue outer measure, and prove that it is, in fact, an outer measure. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet

From playlist Measure Theory

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Outer measures - Part 1 (Measure Theory Part 20)

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From playlist Measure Theory

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Micrometer / diameter of daily used objects

What was the diameter? music: https://www.bensound.com/

From playlist Fine Measurements

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Outer measures - Part 3: Proof (Measure Theory Part 22)

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From playlist Measure Theory

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Outer measures - Part 2: Examples (Measure Theory Part 21)

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From playlist Measure Theory

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From playlist Area and Perimeter

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From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021

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MIT 18.102 Introduction to Functional Analysis, Spring 2021 Instructor: Dr. Casey Rodriguez View the complete course: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ YouTube Playlist: https://www.youtube.com/watch?v=cqdUuREzGuo&list=PLUl4u3cNGP63micsJp_

From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021

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MIT 18.102 Introduction to Functional Analysis, Spring 2021 Instructor: Dr. Casey Rodriguez View the complete course: https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ YouTube Playlist: https://www.youtube.com/watch?v=pWs93gASTJk&list=PLUl4u3cNGP63micsJp_

From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021

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From playlist The New CHALKboard

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English version here: https://youtu.be/LC-9KzxVoWI Abonniert den Kanal oder unterstützt ihn auf Steady: https://steadyhq.com/en/brightsideofmaths Oder unterstützt via PayPal: https://paypal.me/brightmaths Offizielle Unterstützer in diesem Monat: - Ran Gutin - Petar Djurkovic - William Rip

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Tunneling between Landau levels in a quantum dot in the integer and by Marc Röösli

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Ahlfors-Bers 2014 "Teichmüller theory in Outer space"

Mladen Bestvina (University of Utah): I will survey recent progress on the geometry of Outer space, and compare similarities and differences with Teichmüller space.

From playlist The Ahlfors-Bers Colloquium 2014 at Yale

Related pages

Set function | Complete measure | Metric space | Set theory | Non-measurable set | Invariant (mathematics) | Carathéodory's extension theorem | Hausdorff measure | Complement (set theory) | Constantin Carathéodory | Sequence | Inner measure | Empty set | Mathematics | Set (mathematics) | Function (mathematics) | Geometric measure theory | Hausdorff dimension | Σ-algebra | Area | Metric outer measure | Power set