2022-10-02

357: From Convex Open Set Whose Closure Is Bounded on Euclidean Normed C^\infty Manifold into Equal or Higher Dimensional Euclidean Normed C^\infty Manifold Polynomial Map Image of Measure 0 Subset Is Measure 0

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A description/proof of that from convex open set whose closure is bounded on Euclidean normed \(C^\infty\) manifold into equal or higher dimensional Euclidean normed \(C^\infty\) manifold polynomial map image of measure 0 subset is measure 0

Topics


About: Euclidean normed \(C^\infty\) manifold
About: measure
About: map

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any polynomial map from any convex open set whose closure is bounded on any Euclidean normed \(C^\infty\) manifold, into any equal or higher dimensional Euclidean normd \(C^\infty\) manifold, the map image of any measure 0 subset is measure 0.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Description


For any Euclidean normed \(C^\infty\) manifolds, \(\mathbb{R}^n\) and \(\mathbb{R}^m\) where \(n \leq m\), any convex open set whose closure is bounded, \(U \subseteq \mathbb{R}^n\), any polynomial map, \(f: U \rightarrow \mathbb{R}^m\), and any measure \(0\) subset, \(S \subseteq U, m (S) = 0\) where \(m (\bullet)\) is the measure of the argument, the image of \(S\) under \(f\) is measure \(0\), which is \(m (f (S)) = 0\).


2: Proof


The polynomial map can be regarded to be defined on \(\mathbb{R}^n\), canonically extended from \(U\) to \(\mathbb{R}^n\). As \(f\) is \(C^1\), by the proposition that any \(C^1\) map from any open set on any Euclidean normed \(C^\infty\) manifold to any Euclidean normed \(C^\infty\) manifold satisfies the Lipschitz condition in any convex open set whose closure is bounded and contained in the original open set, \(f\) satisfies the Lipschitz condition on \(U\).

By the proposition that the image of any measure 0 subset under any Lipschitz condition satisfying map from any Euclidean normed topological space to any equal or higher dimensional Euclidean normed topological space is measure 0, \(m (f (S)) = 0\).


References


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