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
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.
- Teresa Rajba
- University of Bielsko-Biała, Department of Mathematics and Computer Science, ul. Willowa 2, 43–309 Bielsko-Biała, Poland
- Received: 2010-12-03.
- Revised: 2011-06-03.
- Accepted: 2011-06-03.
