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 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
 December 26th, 2016, 11:58 AM #2 Senior Member   Joined: Aug 2012 Posts: 2,324 Thanks: 715 Step 1. Can you see that the unit circle has measure zero in the plane? December 28th, 2016, 03:14 PM #3 Senior Member   Joined: Oct 2016 From: Arizona Posts: 207 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. 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, 04:36 PM   #4
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Quote:
 Originally Posted by kNiEsSoKk Could someone explain to me why: if M is a k-dim manifold in R^n and k
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. February 4th, 2017, 04:43 PM #5 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 The area of a line in 2 dimensional space is 0. The volume of a curve or plane in 3 dimensional space is 0. Tags kdim, manifold, measure Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Robert Lownds Real Analysis 3 June 18th, 2013 09:21 AM darth_coder Real Analysis 2 October 30th, 2012 08:07 AM hadis Math Events 1 June 8th, 2010 12:50 PM hadis Math Events 0 July 11th, 2009 12:31 AM hadis Math Events 0 July 11th, 2009 12:23 AM

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