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LSpice
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Bochner integral over convex sets lies in the conexconvex set?

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ABIM
ABIM
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Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral $$ \bar{\mu}:=\int_{\omega\in \Omega}\, f(\omega)\, \mu(d\omega). $$ If there is a convex set $C\subseteq E$, such that $f(\Omega)\subseteq C$$f(\Omega)$ is a compact subset of $C$ then, is $\bar{\mu}\in C$?

Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral $$ \bar{\mu}:=\int_{\omega\in \Omega}\, f(\omega)\, \mu(d\omega). $$ If there is a convex set $C\subseteq E$, such that $f(\Omega)\subseteq C$ then is $\bar{\mu}\in C$?

Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral $$ \bar{\mu}:=\int_{\omega\in \Omega}\, f(\omega)\, \mu(d\omega). $$ If there is a convex set $C\subseteq E$, such that $f(\Omega)$ is a compact subset of $C$ then, is $\bar{\mu}\in C$?

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ABIM
ABIM
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Bochner integral over convex sets lies in the conex set?

Let $(\Omega,\Sigma,\mu)$ be a probability space, $E$ be a separable Banach space, $f:\Omega\rightarrow E$ be Borel-measurable and consider the Bochner-integral $$ \bar{\mu}:=\int_{\omega\in \Omega}\, f(\omega)\, \mu(d\omega). $$ If there is a convex set $C\subseteq E$, such that $f(\Omega)\subseteq C$ then is $\bar{\mu}\in C$?