Russian: Нестандартная дискретизация и продолжение по Лебу семейства мер, Сибирский Математический Журнал 34 (1993), no. 3, 190-198.

English translation: Nonstandard discretization and the Loeb extension of a family of measures. Siberian Mathematical Journal 34 (1993), no. 3, 566-573.

Math. Reviews number: 94k:28034
Zentralblatt fur Mathematik number: 808.28014

## Abstract:

Russian: В первой части работы проводится дискретизация интегрального оператора методами нестандартного анализа. Доказывается, что интегральный оператор можно с точностью до бесконечно малого приблизить матрицей бесконечно большого размера. Во второй части строится конструкция меры Леба для случайной меры и рассматривается связь этой конструкции с векторными мерами Леба и нестандартными оболочками.

English: The present article consists of two parts. The first concerns a discretization of an integral operator and uses the discretization of the integral discovered by E.I. Gordon. The main result reads that we can approximate any integral operator to within an infinitesimal by a matrix of infinite size by replacing functions by vectors composed of their values 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 object appears as a result if we consider the random measure as a vector one and construct the corresponding Loeb measure from the vector measure.

Informal: Recall that nonstandard analysis allows the use of infinite (or unlimited) and infinitesimal numbers. Ye.I.Gordon showed that one can choose an unlimited number of points in a measure space, so that the integral of every (standard) function can be evaluated (up to an infinitesimal) as the sum of the values of the function at these points with some fixed weights. Thus, if we substitute a function with the vector of its values at the points, then the integral of the function can be computed (up to an infinitesimal) as the scalar product of the corresponding vector and the (fixed) weights vector.

In the first part of the paper we apply similar approach to integral operators. Precisely, we show that every integral (or even pseudointegral) operator can be viewed as a matrix (of unlimited size), and the action of the operator on a function can be evaluated (up to an infinitesimal) as the action of this matrix on the corresponding vector.

In the second part of the paper we generalize Loeb measure construction to random measures. Recall, that the Loeb construction uses nonstandard analysis to produce a "nice" measure from any given (not necessarily standard) measure. The constraction is more or less straightforward, except the proof that the obtained object is again a random measure.