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

• Main Author: Teresa Rajba
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2012-01-01
• Series: Opuscula Mathematica
• Subjects: backward translation operator; backward difference operator; integral inequality; extreme point
• Online Access: http://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3223.pdf

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 \(\Theta\).