Real Analysis: Are lebesgue measures metrics?

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Homework Help Overview

The discussion revolves around whether Lebesgue measures can be classified as metrics within the context of real analysis. The original poster is exploring the relationship between metrics and Lebesgue measures while learning about Lebesgue Integration.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the properties that define metrics and measures, questioning if Lebesgue measures, which represent length, area, and volume, can be considered metrics. Some participants clarify that metrics measure distances between points, while measures assess the size of sets.

Discussion Status

The discussion is ongoing, with participants providing insights into the definitions and distinctions between metrics and measures. There is an exchange of ideas, but no explicit consensus has been reached regarding the classification of Lebesgue measures.

Contextual Notes

The original poster is navigating the concepts of metrics and measures while adhering to the constraints of their current understanding and coursework in real analysis.

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Homework Statement


Are Lebesgue measures considered metrics?

The Attempt at a Solution


I have an elementary understanding of metrics and am currently learning about Lebesgue Integration. A Lebesgue measure as a length in Euclidean space should be a metric, but what about areas and volumes? I'm thinking no because metrics are distances, which to me means length. Is there a way I can prove this? Are there necessary properties of each that agree or disagree?
 
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A "metric" is a way of measuring the size of a set, not just the "length" of an interval. We use "metrics" in 2 and 3 dimensions all the time!
 
No. Because, as you said, a metric measures the distance between two things. A 'measure' (in the Lebesgue measure sense) measures the size of one thing. How can one be the other? The number of 'things' is different.
 
Oh, I see your point. Thanks.
 

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