Titlebook: Measure and Category; A Survey of the Anal John C. Oxtoby Textbook 1980Latest edition Springer-Verlag New York Inc. 1980 Kategorie (Math.).

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書(shū)目名稱(chēng)Measure and Category影響因子(影響力)




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書(shū)目名稱(chēng)Measure and Category讀者反饋




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entrance

Liouville Numbers,erty in question, in fact, most points of the interval (in the sense of cardinal number, or measure, or category, respectively) have the property. As a first illustration of this method let us consider the existence of transcendental numbers.

無(wú)底

Lebesgue Measure in ,-Space,of the notion of volume to a larger class of sets. Thus Lebesgue measure has a different meaning in spaces of different dimension. However, since we shall usually regard the dimension as fixed, there is no need to indicate r explicitly in our notations.

慢跑鞋

The Property of Baire,e addition and multiplication, respectively. Such a class is also closed under the operations of union and difference. It is therefore a ring of subsets of its union, as this term was defined in Chapter 3.

梯田

Robust

不合

The Banach Category Theorem, union of any family of open sets of measure zero has measure zero (for any measure defined for all open sets). It is remarkable that the first statement remains valid whether the space has a countable base or not. The second statement, however, needs to be qualified.

莊嚴(yán)

,The Poincaré Recurrence Theorem,oint of view, this “recurrence theorem” has a special interest, because in proving it Poincaré anticipated the notions of both measure and category. Publication of his treatise, “Les méthodes nouvelles de la mécanique céleste” [29], antedated slightly the introduction of either notion.

用樹(shù)皮

Non-Measurable Sets,e consistent among themselves. No actual example of anon-measurable set that admits such a representation is known (but see [40, p. 17]). Nevertheless, with the aid of the axiom of choice it is easy to show that non-measurable sets exist. We shall consider several such constructions.

inundate

,The Sierpinski-Erd?s Duality Theorem,to look for a category analogue, or a measure analogue, has very often proved to be a useful guide. In this and the following chapters we shall take a closer look at the duality we have observed between measure and category, to see how far it extends in the case of the line and other spaces, and to discover what underlies it.