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).
Micrometer/diameter of daily used objects.
What was the diameter? music: https://www.bensound.com/
From playlist Fine Measurements
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
Outer measures - Part 1 (Measure Theory Part 20)
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From playlist Measure Theory
Micrometer / diameter of daily used objects
What was the diameter? music: https://www.bensound.com/
From playlist Fine Measurements
Outer measures - Part 3: Proof (Measure Theory Part 22)
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From playlist Measure Theory
Outer measures - Part 2: Examples (Measure Theory Part 21)
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From playlist Measure Theory
What is the formula for the area of a rectangle
👉 Learn how to find the area and perimeter of a rectangle. A rectangle is a parallelogram with all the angles equal to 90 degrees. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. The
From playlist Area and Perimeter
What are the formulas for the measure of angles outside of a circle
Learn the essential definitions of the parts of a circle. A secant line to a circle is a line that crosses exactly two points on the circle while a tangent line to a circle is a line that touches exactly one point on the circle. A chord is a line that has its two endpoints on the circle.
From playlist Essential Definitions for Circles #Circles
How to use dialations and factors to find the new perimeter and area of a rectangle
👉 Learn how to find the area and perimeter of a rectangle. A rectangle is a parallelogram with all the angles equal to 90 degrees. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. The
From playlist Area and Perimeter
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=OHiu2F18dFA&list=PLUl4u3cNGP63micsJp_
From playlist MIT 18.102 Introduction to Functional Analysis, Spring 2021
Lecture 8: Lebesgue Measurable Subsets and Measure
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
Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers
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
Some Small Ideas in Math: A Set of Measure Zero Versus a Set of First Category (Meager Sets)
There are a ton of different ways to define what it means for a set to be "small". Here, we will focusing on the difference between a set of measure zero versus a set of first category by using examples to demonstrate that they are different sizing methods. Depending on the context of the
From playlist The New CHALKboard
MIT 9.04 Sensory Systems, Fall 2013 View the complete course: http://ocw.mit.edu/9-04F13 Instructor: Chris Brown This video describes the anatomy of the cochlea, inner and outer hair cells as well as OHC electromotility and otoacoustic emissions. License: Creative Commons BY-NC-SA More i
From playlist MIT 9.04 Sensory Systems, Fall 2013
Real Analysis - Eva Sincich - Lecture 01
From playlist Machine learning
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From playlist Maßtheorie und Integrationstheorie
Tunneling between Landau levels in a quantum dot in the integer and by Marc Röösli
DISCUSSION MEETING : EDGE DYNAMICS IN TOPOLOGICAL PHASES ORGANIZERS : Subhro Bhattacharjee, Yuval Gefen, Ganpathy Murthy and Sumathi Rao DATE & TIME : 10 June 2019 to 14 June 2019 VENUE : Madhava Lecture Hall, ICTS Bangalore Topological phases of matter have been at the forefront of r
From playlist Edge dynamics in topological phases 2019
Finding the area and perimeter of a rectangle
👉 Learn how to find the area and perimeter of a rectangle. A rectangle is a parallelogram with all the angles equal to 90 degrees. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. The
From playlist Area and Perimeter
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