Given a (finitely additive) full conditional probability space $(X,\F\times\F^0,\mu)$ and a conditional measurable space $(Y,\G\times\G^0)$, a multivalued mapping $\Gamma$ from $X$ to $Y$ induces a class of full conditional probabilities on $(Y,\G\times\G^0)$. A closed form expression for the lower and upper envelopes $\mu_*$ and $\mu^*$ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures. For every $B \in \G^0$, $\mu_*(\cdot|B)$ is a normalized totally monotone capacity which is continuous from above if $(X,\F\times\F^0,\mu)$ is a countably additive full conditional probability space and $\F$ is a $\sigma$-algebra. Moreover, the full conditional prevision functional $\MM$ induced by $\mu$ on the set of $\F$-continuous conditional gambles is shown to give rise through $\Gamma$ to the lower and upper full conditional prevision functionals $\MM_*$ and $\MM^*$ on the set of $\G$-continuous conditional gambles. For every $B \in \G^0$, $\MM_*(\cdot|B)$ is a totally monotone functional having a Choquet integral expression involving $\mu_*$. Finally, by considering another conditional measurable space $(Z,\H\times\H^0)$ and a multivalued mapping from $Y$ to $Z$, it is shown that the conditional measures $\mu_**$, $\mu^**$ and functionals $\MM_**$, $\MM^**$ induced by $\mu_*$ preserve the same properties of $\mu_*, \mu^*$ and $\MM_*$, $\MM^*$.

Upper and lower conditional probabilities induced by a multivalued mapping

PETTURITI, DAVIDE;
2018

Abstract

Given a (finitely additive) full conditional probability space $(X,\F\times\F^0,\mu)$ and a conditional measurable space $(Y,\G\times\G^0)$, a multivalued mapping $\Gamma$ from $X$ to $Y$ induces a class of full conditional probabilities on $(Y,\G\times\G^0)$. A closed form expression for the lower and upper envelopes $\mu_*$ and $\mu^*$ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures. For every $B \in \G^0$, $\mu_*(\cdot|B)$ is a normalized totally monotone capacity which is continuous from above if $(X,\F\times\F^0,\mu)$ is a countably additive full conditional probability space and $\F$ is a $\sigma$-algebra. Moreover, the full conditional prevision functional $\MM$ induced by $\mu$ on the set of $\F$-continuous conditional gambles is shown to give rise through $\Gamma$ to the lower and upper full conditional prevision functionals $\MM_*$ and $\MM^*$ on the set of $\G$-continuous conditional gambles. For every $B \in \G^0$, $\MM_*(\cdot|B)$ is a totally monotone functional having a Choquet integral expression involving $\mu_*$. Finally, by considering another conditional measurable space $(Z,\H\times\H^0)$ and a multivalued mapping from $Y$ to $Z$, it is shown that the conditional measures $\mu_**$, $\mu^**$ and functionals $\MM_**$, $\MM^**$ induced by $\mu_*$ preserve the same properties of $\mu_*, \mu^*$ and $\MM_*$, $\MM^*$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/1417277
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