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Nonstandard discretization and the Loeb extension of a family of measures. (English. Russian original) Zbl 0808.28014
Sib. Math. J. 34, No. 3, 566-573 (1993); translation from Sib. Mat. Zh. 34, No. 3, 190-198 (1993).
The present article consists of two parts. The first concerns a discretization of an integral operator and uses the discretization of the integral which was discovered by E. I. Gordon. The main result asserts that it is possible to approximate any integral operator (within infinitesimals) by a matrix of infinite size by replacing functions with vectors composed of the values of the functions at a finite (but unlimited) number of points. In the second part, we implement the Loeb construction for a random measure. We prove that the same results if we treat the random measure as a vector measure and construct the corresponding Loeb measure from the latter.
We use the language of the Kawai theory NST; however, our reasoning remains correct within the classical Robinson nonstandard analysis.
28E05 Nonstandard measure theory