Null subset of a differential manifold - Revision history

Vipul: 6 revisions

2008-05-18T19:50:18Z

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Vipul: /* Definition */

2008-01-13T20:39:21Z

Definition

← Older revision		Revision as of 20:39, 13 January 2008
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	Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> is said to have '''measure zero''' or to be a '''null subset''' if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.		Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> is said to have '''measure zero''' or to be a '''null subset''' if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.

	The well-definedness of this notion rests on the fact that any diffeomorphism ~~from~~ <math>\R^n</math> ~~to itself~~ maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]). {{further\|[[Diffeomorphism implies nullset-preserving]]}}		The well-definedness of this notion rests on the fact that any diffeomorphism between open subsets of <math>\R^n</math> maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]). {{further\|[[Diffeomorphism implies nullset-preserving]]}}

	==Facts==		==Facts==

Vipul at 20:38, 13 January 2008

2008-01-13T20:38:57Z

← Older revision		Revision as of 20:38, 13 January 2008
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	==Definition==		==Definition==

	Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> ~~has~~ measure zero if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.		Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> is said to have '''measure zero''' or to be a '''null subset''' if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.

	The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]).		The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]). {{further\|[[Diffeomorphism implies nullset-preserving]]}}

	==Facts==		==Facts==

Vipul: Measure zero subset of a differential manifold moved to Null subset of a differential manifold

2008-01-13T20:37:47Z

Measure zero subset of a differential manifold moved to Null subset of a differential manifold

← Older revision	Revision as of 20:37, 13 January 2008
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Vipul at 00:33, 13 January 2008

2008-01-13T00:33:18Z

← Older revision		Revision as of 00:33, 13 January 2008
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	The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]).		The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]).

	~~The most significant occurrence~~ of measure zero ~~subsets is in~~ [[Sard's theorem]].		==Facts==

			* Any [[submanifold]] of codimension at least 1, is a measure zero subset
			* [[Sard's theorem]] is a generalization of the above

Vipul at 00:32, 13 January 2008

2008-01-13T00:32:04Z

← Older revision		Revision as of 00:32, 13 January 2008
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	Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> has measure zero if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.		Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> has measure zero if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.

	The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[manifold]]).		The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[topological manifold]]).

			The most significant occurrence of measure zero subsets is in [[Sard's theorem]].

Vipul at 00:31, 13 January 2008

2008-01-13T00:31:29Z

New page

==Definition==

Let <math>M</math> be a [[differential manifold]]. A subset <math>S</math> of <math>M</math> has measure zero if the following holds: for any open subset <math>U</math> of <math>M</math> and any [[diffeomorphism]] between <math>U</math> and <math>\R^n</math>, the image of <math>U \cap S</math> under the diffeomorphism has measure zero in <math>\R^n</math>.

The well-definedness of this notion rests on the fact that any diffeomorphism from <math>\R^n</math> to itself maps measure zero subsets to measure zero subsets (this is ''not'' true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does ''not'' make sense for a [[manifold]]).