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 $\MM_*$, $\MM^*$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
