# Integration with filters

@inproceedings{Bottazzi2020IntegrationWF, title={Integration with filters}, author={Emanuele Bottazzi and Monroe Eskew}, year={2020} }

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral…

## References

SHOWING 1-10 OF 50 REFERENCES

Extensions of Measure.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1938

Introduction.-One of the elementary applications of measure theory is, using two dimensional terminology, the determination of areas in the Euclidean plane. Initially, area is defined only for…

MEASURE THEORY AND INTEGRATION ON THE LEVI-CIVITA FIELD

- Mathematics
- 2003

It is well known that the disconnectedness of a non-Archimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy…

Conversion from nonstandard to standard measure spaces and applications in probability theory

- Mathematics
- 1975

Let (X, (!, v) be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard o'algebra Xi containing the…

Nonstandard measure theory–Hausdorff measure

- Mathematics
- 1977

In this paper it is shown that the Hausdorff measures A' for t E [0, oo) can be simultaneously represented as *finite counting measures in an appropriate nonstandard model. That is, the following…

On the nonstandard representation of measures

- Mathematics
- 1972

In this paper it is shown that every finitely additive probability measure ,u on S which assigns 0 to finite sets can be given a nonstandard representation using the counting measure for some…

Elementary numerosity and measures

- MathematicsJ. Log. Anal.
- 2014

The notion of elementary numerosity as a special function dened on all subsets of a given set which takes values in a suitable non-Archimedean field, and satises the same formal properties of finite cardinality is introduced.

A non-standard representation for Brownian Motion and Itô integration

- Mathematics
- 1976

In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and…

An arithmetic property of Riemann sums

- Mathematics, Philosophy
- 1964

holds for all real x if / is Riemann integrable on [0, l]. In the present note it is shown that there are bounded measurable functions / for which (2) is false for every x and that this convergence…

Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations

- Mathematics, PhysicsAdvances in Mathematics
- 2019