Abstract
The paper pursues the definition of a maxitive integral on all real-valued functions (i.e., the integral of the pointwise maximum of two functions must be the maximum of their integrals). This definition is not determined by maxitivity alone: additional requirements on the integral are necessary. The paper studies the consequences of additional requirements of invariance with respect to affine transformations of the real line.
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References
de Finetti, B.: Theory of Probability, 2 vols. Wiley (1974-1975)
Savage, L.J.: The Foundations of Statistics, 2nd edn. Dover (1972)
Kolmogorov, A.N.: Foundations of the Theory of Probability, 2nd edn. Chelsea Publishing (1956)
Solovay, R.M.: A model of set-theory in which every set of reals is Lebesgue measurable. Ann. Math. (2) 92, 1–56 (1970)
Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)
Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and Its Applications. Springer (1997)
Dubois, D., Prade, H.: Possibility Theory. Plenum Press (1988)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)
de Cooman, G., Aeyels, D.: Supremum preserving upper probabilities. Inf. Sci. 118, 173–212 (1999)
Shackle, G.L.S.: Expectation in Economics. Cambridge University Press (1949)
Cohen, L.J.: A logic for evidential support. Br. J. Philos. Sci. 17, 21–43, 105–126 (1966)
Neyman, J., Pearson, E.S.: On the use and interpretation of certain test criteria for purposes of statistical inference. Biometrika 20A, 175–240, 263–294 (1928)
Cattaneo, M.: Likelihood decision functions. Electron. J. Stat. 7, 2924–2946 (2013)
Cattaneo, M.: On maxitive integration. Technical Report 147, Department of Statistics, LMU Munich (2013)
Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of Charges. Academic Press (1983)
Ash, R.B.: Real Analysis and Probability. Academic Press (1972)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn. Birkhäuser (2009)
Sugeno, M.: Theory of Fuzzy Integrals and Its Applications. PhD thesis, Tokyo Institute of Technology (1974)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)
Denneberg, D.: Non-Additive Measure and Integral. Kluwer (1994)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)
Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)
Föllmer, H., Schied, A.: Stochastic Finance, 3rd edn. De Gruyter (2011)
Yoo, K.R.: The iterative law of expectation and non-additive probability measure. Econ. Lett. 37, 145–149 (1991)
Giang, P.H., Shenoy, P.P.: Decision making on the sole basis of statistical likelihood. Artif. Intell. 165, 137–163 (2005)
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Cattaneo, M.E.G.V. (2014). Maxitive Integral of Real-Valued Functions. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 442. Springer, Cham. https://doi.org/10.1007/978-3-319-08795-5_24
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DOI: https://doi.org/10.1007/978-3-319-08795-5_24
Publisher Name: Springer, Cham
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