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December 26th, 2016, 01:21 AM  #1 
Newbie Joined: Oct 2016 From: Netherlands Posts: 3 Thanks: 0  kdim manifold, measure 0
Could someone explain to me why: if M is a kdim manifold in R^n and k<n , then M has measure 0. 
December 26th, 2016, 12:58 PM  #2 
Senior Member Joined: Aug 2012 Posts: 951 Thanks: 185 
Step 1. Can you see that the unit circle has measure zero in the plane?

December 28th, 2016, 04:14 PM  #3 
Member Joined: Oct 2016 From: New York Posts: 49 Thanks: 13 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.

December 28th, 2016, 05:36 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 114 Thanks: 44 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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.  
February 4th, 2017, 05:43 PM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,279 Thanks: 570 
The area of a line in 2 dimensional space is 0. The volume of a curve or plane in 3 dimensional space is 0. 

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kdim, manifold, measure 
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