On sigma-ideals of sets

Author:
C. G. Mendez

Journal:
Proc. Amer. Math. Soc. **60** (1976), 124-128

MSC:
Primary 28A05; Secondary 04A15

DOI:
https://doi.org/10.1090/S0002-9939-1976-0417359-8

MathSciNet review:
0417359

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Abstract: Let denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. (i) *and* *are* -*ideals*. (ii) *The union of* *or* *is* . (iii) *The complement of each member of* *or* *contains a set of power c belonging to* *and* , *respectively*, (iv) *The unit square may be represented as the union of two complementary Borel sets*: *one in* *and* *and the other one of Lebesgue measure zero and first category*, (v) *The unit square may be represented as the union of two complementary Borel sets*: *one in* *and the other one in* . Theorem 2. *does not satisfy* (vi) *There is a subclass* *of power* *c of the class* *such that every member of the class is contained in some member of the subclass*. Theorem 3. *There does not exist a one-to-one mapping f from* *onto itself, such that* *is a Lebesgue measure zero* (*first category*) *subset of* . Theorems 2 and 3 hold for more general .

A theorem on the theory of quotient (Boolean) algebras follows from these results.

**[1]**A. Horn and A. Tarski,*Measures in Boolean algebras*, Trans. Amer. Math. Soc.**64**(1948), 467-497. MR**10**, 518. MR**0028922 (10:518h)****[2]**C. G. Mendez,*The analogy between the concepts of first category and measure zero*, M.A. Thesis, Univ. of Colorado, 1968.**[3]**-,*On Borel and analytic structures and the interrelation of measure and category in an abstract setting. On sigma-ideals of sets and on measures in*, Ph.D. Dissertation, Univ. of Colorado, 1974.**[4]**J. C. Oxtoby,*Measure and category*, Springer-Verlag, New York, 1971. MR**584443 (81j:28003)****[5]**R. Sikorski,*On the inducing of homomorphisms by mappings*, Fund. Math.**36**(1949), 7-22. MR**11**, 166. MR**0031535 (11:166a)****[6]**-,*Boolean algebras*, 2nd ed., Academic Press, New York, 1964. MR**31**#2178. MR**0177920 (31:2178)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0417359-8

Keywords:
Measure zero,
first category,
-ideals of sets,
set theoretically equivalent classes of sets,
continuum,
Sierpiński-Erdös duality theorem,
quotient (Boolean) algebras

Article copyright:
© Copyright 1976
American Mathematical Society