Real Analysis: Are lebesgue measures metrics?

[binks01]

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?

 

[HallsofIvy]

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!

 

[Dick]

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.

 

[HallsofIvy]

Oh, I see your point. Thanks.