Opuscula Math. 32, no. 2 (2012), 317-326
http://dx.doi.org/10.7494/OpMath.2012.32.2.317

Opuscula Mathematica

# An application of the Choquet theorem to the study of randomly-superinvariant measures

Teresa Rajba

Abstract. Given a real valued random variable $$\Theta$$ we consider Borel measures $$\mu$$ on $$\mathcal{B}(\mathbb{R})$$, which satisfy the inequality $$\mu(B) \geq E\mu(B-\Theta)$$ ($$B \in \mathcal{B}(\mathbb{R})$$) (or the integral inequality $$\mu(B) \geq \int_{-\infty}^{+\infty} \mu(B-h)\gamma (dh)$$). We apply the Choquet theorem to obtain an integral representation of measures $$\mu$$ satisfying this inequality. We give integral representations of these measures in the particular cases of the random variable $$\Theta$$.

Keywords: backward translation operator, backward difference operator, integral inequality, extreme point.

Mathematics Subject Classification: 60E15, 26D10.

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• Teresa Rajba
• University of Bielsko-Biała, Department of Mathematics and Computer Science, ul. Willowa 2, 43–309 Bielsko-Biała, Poland
• Revised: 2011-06-03.
• Accepted: 2011-06-03.