Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-30T21:22:34.282Z Has data issue: false hasContentIssue false

1 - Lebesgue Measure

from PART I - Measure And Integration

Published online by Cambridge University Press:  06 January 2022

Martin Buntinas
Martin Buntinas
Affiliation:
Loyola University, Chicago
Get access

Summary

Outer, Inner, and Lebesgue Measure are defined and systematically studied; first for (n-dimensional) intervals, then for finite and countable union of intervals, then for open and closed sets, and finally for general Lebesgue Measurable sets in Euclidean Spaces. The Approximation Theorem and the Caratheodory Characterization of Measurability are proven. Borel sets are studied and examples are given of Nonmeasurable Sets, as well as Measurable Sets which are not Borel.

Keywords

Outer MeasureInner MeasureLebesgue MeasureApproximation TheoremCaratheodory CharacterizationBorel SetsNonmeasurable Sets
Type
Chapter
Information
Classical and Discrete Functional Analysis with Measure Theory , pp. 65 - 96
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Lebesgue Measure
  • Martin Buntinas, Loyola University, Chicago
  • Book: Classical and Discrete Functional Analysis with Measure Theory
  • Online publication: 06 January 2022
  • Chapter DOI: https://doi.org/10.1017/9781139524445.004
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Lebesgue Measure
  • Martin Buntinas, Loyola University, Chicago
  • Book: Classical and Discrete Functional Analysis with Measure Theory
  • Online publication: 06 January 2022
  • Chapter DOI: https://doi.org/10.1017/9781139524445.004
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Lebesgue Measure
  • Martin Buntinas, Loyola University, Chicago
  • Book: Classical and Discrete Functional Analysis with Measure Theory
  • Online publication: 06 January 2022
  • Chapter DOI: https://doi.org/10.1017/9781139524445.004
Available formats
×