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# Elements of measure theory

The Lebesgue measure, being a function from family of subsets of into , is not of the form , so it does not lie directly in a scope of this article. However, it is certainly one of the main tools of real analysis and many results concerning its generalizations have a deep set theoretical context. Therefore, a short section concerning the newest developments in this area has been added to this paper.

An accessible survey concerning different extensions of Lebesgue measure can be found in the 1989 Mathematical Intelligencer article [22] of K. Ciesielski. The best survey concerning universal (i.e., defined on ) countable additive extensions of Lebesgue measure can be found in the 1993 survey article of D. H. Fremlin [58]. Thus, we will concentrate here only on the newest developments, that concern isometrically invariant extensions of Lebesgue measure. (See also M. Laczkovich survey article [90] on this subject.)

Recall here, that by the 1923 theorem of Banach there is a finitely additive isometrically invariant measure extending Lebesgue measure, while such a measure on does not exist by a famous Banach-Tarski Paradox (1924):

• the ball and the cube (of arbitrary volumes) are isometrically equivalent, i.e., there is a finite partition of B and isometries of such that forms a partition of Q.
There were two famous problems around this subject. The first one, due to Marczewski, was whether the pieces in the Banach-Tarski Paradox can have the Baire property. The answer to this question, surprisingly positive, was obtained by Dougherty and Foreman in 1994.

The second famous question was the Tarski's circle-squaring problem: is a circle of the unit area equivalent to a square of the unit area? Note that if the areas of C and D were different, then Banach's theorem of 1923 would immediately imply the negative answer. However, the answer to Tarski's circle-squaring problem is positive, as proved by Laczkovich in 1990.

The other class of isometrically invariant extensions of Lebesgue measures on concerns countably additive extensions. In 1936 Sierpinski asked, whether such an extension can be maximal. The negative answer to this question was given in 1977 by A. B. Kharazishvili [77] (for n=1) and in 1985 by Ciesielski and Pelc [40] (for arbitrary n). (Compare also [23, 24, 85, 143].)

A weak side of this theorem was that the extension of was only by (new) measure zero sets so, in a way, trivial. This has been recently improved by Zakrzewski, who showed

Many interesting results concerning isometrically invariant extensions of Lebesgue measure can be also found in 1983 book of Kharazishvili [78].

Next: References Up: Set Theoretic Real Analysis Previous: Some elements of topology