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Abdelmalek Abdesselam
Abdelmalek Abdesselam
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Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$$$ f \colon \Omega\to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \Omega\to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

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Gerardo Mastracchio
Gerardo Mastracchio
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Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$ is measurable. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$ is measurable, where $C(K,E)$ is the space of continuous functions on $K$ to $E$, with the supremum norm. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks

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Gerardo Mastracchio
Gerardo Mastracchio
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Measurability of random function with values in $C(K,E)$

Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function $$ f \colon \to C(K,E) $$ is measurable. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started?

Thanks