My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
December 26th, 2016, 12:21 AM   #1
Newbie
 
Joined: Oct 2016
From: Netherlands

Posts: 3
Thanks: 0

k-dim manifold, measure 0

Could someone explain to me why:

if M is a k-dim manifold in R^n and k<n , then M has measure 0.
kNiEsSoKk is offline  
 
December 26th, 2016, 11:58 AM   #2
Senior Member
 
Joined: Aug 2012

Posts: 1,574
Thanks: 380

Step 1. Can you see that the unit circle has measure zero in the plane?
Maschke is offline  
December 28th, 2016, 03:14 PM   #3
Member
 
ProofOfALifetime's Avatar
 
Joined: Oct 2016
From: New York

Posts: 56
Thanks: 14

Math Focus: Analysis and Differential Geometry
I can't wait until I'm at this level of math! I can't wait to learn measure theory! The end of Rudin's book starts it, but I'm only on Chapter 2 so far.
ProofOfALifetime is offline  
December 28th, 2016, 04:36 PM   #4
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 167
Thanks: 79

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by kNiEsSoKk View Post
Could someone explain to me why:

if M is a k-dim manifold in R^n and k<n , then M has measure 0.
The proof of this is not exactly easy. However, the intuition is pretty straight forward. It follows these ideas.

1. One sufficient condition for a set to have measure (with respect to $n$ dimensional Lebesgue measure) zero is that you can cover it by a set of arbitrarily small measure. Convince yourself that any $k$ dimesional affine subset of $\mathbb{R}^n$ must have measure zero.

2. An intuitive way of thinking about a manifold is as "locally" Euclidean subset. This means that for any point there is a neighborhood on which the topology is locally homeomorphic to a ball in $\mathbb{R}^n$.

Combining the above 2 ideas should hopefully give you an intuitive idea about why the theorem must be true.
SDK is offline  
February 4th, 2017, 04:43 PM   #5
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 2,734
Thanks: 707

The area of a line in 2 dimensional space is 0.

The volume of a curve or plane in 3 dimensional space is 0.
Country Boy is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
kdim, manifold, measure



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Parametrization of a manifold Robert Lownds Real Analysis 3 June 18th, 2013 09:21 AM
Manifold darth_coder Real Analysis 2 October 30th, 2012 08:07 AM
manifold hadis Math Events 1 June 8th, 2010 12:50 PM
manifold hadis Math Events 0 July 11th, 2009 12:31 AM
manifold hadis Math Events 0 July 11th, 2009 12:23 AM





Copyright © 2017 My Math Forum. All rights reserved.