Nearstandardness of 
finite set

T. Kudryk; V. Lyantse; G. Chuiko

On the set ${\Bbb T}$, finite in the sense of the E.Nelson's Internal Set Theory, a measure $\nu$ is given. For ''discrete integral'' $\Sigma_{t\in{\Bbb T}}x(t)\nu\{t\},\,x\in{\Bbb C}^{\Bbb T}$ the analogs of the classical theorems of Lebesgue integral theory are regarded. As the set ${\Bbb T}$ is nonstandard, for the measures and functions given on it, a direct definition of nearstandardness is impossible. Indirect way based on the embedding of the algebra $2^{\Bbb T}$ into the algebra $2^{\text{\bf T}}$ is used. Here, {\bf T} is a standard set. Relations between nearstandard charges and their shadows, as well as nearstandard functions and their shadows appearing with such approach are investigated.

1. Albeverio S., Fenstad J., Hoegh-Krohn T., Lindstrom T. Nonstandard methods in stochastic analysis and mathematical physics. – New York: Academic Press, 1986.

2. Cutland N.J. Nonstandard measure theory and its applications // Bull. London Math. Soc. 1983. V. 15, N 6. P. 529–589.

3. Loeb P.A. Conversion from nonstandard to standard measure spaces and applications in probability theory // Trans. Amer. Math. Soc. 1975. V. 211. P. 113–122.

4. Lutz R., Goze M. Nonstandard analysis: a practical guide with applications. –Berlin: Lecture Notes in Mathematics, 881. 1981. 261 p

5. Nelson E. Internal set theory: a new approach to nonstandard analysis // Bull. Amer. Math. Soc. 1977. V. 83, N 6. P. 1165–1198.

6. Nelson E. Radically elementary probability theory – Princeton, New Jersey: Princeton University Press, 1987. 97 p Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine

	Nearstandardness of finite set
Author	T. Kudryk, V. Lyantse, G. Chuiko Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine
Abstract	On the set ${\Bbb T}$, finite in the sense of the E.Nelson's Internal Set Theory, a measure $\nu$ is given. For ''discrete integral'' $\Sigma_{t\in{\Bbb T}}x(t)\nu\{t\},\,x\in{\Bbb C}^{\Bbb T}$ the analogs of the classical theorems of Lebesgue integral theory are regarded. As the set ${\Bbb T}$ is nonstandard, for the measures and functions given on it, a direct definition of nearstandardness is impossible. Indirect way based on the embedding of the algebra $2^{\Bbb T}$ into the algebra $2^{\text{\bf T}}$ is used. Here, {\bf T} is a standard set. Relations between nearstandard charges and their shadows, as well as nearstandard functions and their shadows appearing with such approach are investigated.
Keywords	Internal Set Theory, finite set, measure, discrete integral, Lebesgue integral, nonstandard analysis, nearstandardness
DOI	doi:10.30970/ms.2.1.25-34
Reference	1. Albeverio S., Fenstad J., Hoegh-Krohn T., Lindstrom T. Nonstandard methods in stochastic analysis and mathematical physics. – New York: Academic Press, 1986. 2. Cutland N.J. Nonstandard measure theory and its applications // Bull. London Math. Soc. 1983. V. 15, N 6. P. 529–589. 3. Loeb P.A. Conversion from nonstandard to standard measure spaces and applications in probability theory // Trans. Amer. Math. Soc. 1975. V. 211. P. 113–122. 4. Lutz R., Goze M. Nonstandard analysis: a practical guide with applications. –Berlin: Lecture Notes in Mathematics, 881. 1981. 261 p 5. Nelson E. Internal set theory: a new approach to nonstandard analysis // Bull. Amer. Math. Soc. 1977. V. 83, N 6. P. 1165–1198. 6. Nelson E. Radically elementary probability theory – Princeton, New Jersey: Princeton University Press, 1987. 97 p Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine
Pages	25-34
Volume	2
Issue	1
Year	1993
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Nearstandardness of finite set