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- Dictionarymeasure/ˈmɛʒə/
verb
- 1. ascertain the size, amount, or degree of (something) by using an instrument or device marked in standard units: "the amount of water collected is measured in pints" Similar Opposite
- 2. assess the importance, effect, or value of (something): "it is hard to measure teaching ability" Similar
noun
- 1. a plan or course of action taken to achieve a particular purpose: "cost-cutting measures" Similar
- 2. a standard unit used to express the size, amount, or degree of something: "a furlong is an obsolete measure of length" Similar
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A classical solution to the measure problem consists in attempting to approximate the measure of a complicated set using simple sets. More precisely, suppose we have a class of simple sets S which we know how to measure (these would contain rectangles and finite unions of rectangle for example).
14. Having measure zero is something that can be defined without actually defining a measure on the manifold. A subset A A of N N can be defined to have measure zero if for every chart ϕ: U → Rn ϕ: U → R n of N N, ϕ(U ∩ A) ϕ (U ∩ A) has measure zero in Rn R n. You would want this to be something that is invariant under diffeomorphism.
Oct 10, 2022 · How would you define a "relatively natural" measure on a manifold without smooth structure? I have read here that the "standard" way to find a Lebesgue measure on a smooth (compact? orientable?) manifold is to choose a Riemannian metric. This gives you a volume form on each chart and the coordinate changes are compatible with these local volume forms. You can then define a linear form on continuous functions which allows you to use the Riesz representation theorem. What can you do if you don ...
An outer measure is a set function defined on some set X with the properties that empty returns 0, of sub additivity and of monotonicity. From this outer measure we get a sigma algebra of measurable sets. On this sigma algebra the outer measure is a measure with the property of additivity for unions of disjoint sets.
Apr 28, 2023 · The intuitive motivation for this question is to consider the "probability" of a given vibration mode to be contained in a "random" perturbation. My questions is that, given an L2 L 2 space, is it possible to define a measure on it in order to make sense of some "probability"?
There is a theory of "metric measure spaces" which are metric spaces with a Borel measures, ie., a measure compatible with the topology of the metric space. It has a big literature that is well represented online. There is also the trivial answer, that if you don't demand compatibility of the measure and the metric, there is no particular relation between them. A wider neighborhood has bigger measure, and easy things like that, but not necessarily anything nontrivial.
Feb 9, 2022 · Is this the right way to construct the canonical measure on the Borel σ σ -algebra of an oriented Riemannian manifold? If so, is this the most commonly used measure for geometric analysis (e.g. Jost's Riemannian Geometry and Geometric Analysis)?
Nov 25, 2022 · Parting again from the Lebesgue outer measure μ∗, one defines Lebesgue measurable sets as those which can be 'approximated from above' by open sets arbitrarily well: Definition: a set E ⊆ Rd is Lebesgue measurable iff for every ε> 0 there is some open set U ⊆Rd containing E such that μ∗(U ∖ E) <ε. Then the Lebesgue measure is characterized as the restriction of μ∗ to Lebesgue measurable sets, and a bit of work shows that the collection of Lebesgue measurable sets is a σ ...
Then, is this measure uniquely determined? I know if I tell you how to integrate all measurable functions, then this measure is of course uniquely determined. Because integrate characteristic functions will give you measure of that respective set. But is it also true if I only define integration with continuous functions?
May 6, 2019 · According to your definition measure depends strongly on parametrization i.e. if you choose another parametrization ϕ(x) = φ(λx) for some real λ the measure would be multiplied by λ too. Let ι: M → Rn be inclusion map. It induces differentials on tangent spaces, which can be used to define Riemann metric on M: if v, w ∈ TpM, g(v, w ...
