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Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: herehere]

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: here]

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: here]

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Nduccio
Nduccio
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Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: here]

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]

[Linked thread on MSE: here]

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Nduccio
Nduccio
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Measurable sets of probability measures $\{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}$

Let $(X,\mathscr{F})$ be a measurable space, and let $M$ be the set all probability measures $\mu: \mathscr{F} \to [0,1]$. Let us denote with $\mathscr{M}$ the $\sigma$-algebra on $M$ generated by the mappings $\mu \mapsto \mu(F)$, with $F \in \mathscr{F}$.

Now fix a Borel set $B$ of $\mathbb{R}$ and $A \in \mathscr{F} \otimes \mathscr{F}$. How can we prove that $$ \{\mu \in M: (\mu \times \mu)(A) \in B\} \in \mathscr{M}? $$

[Here $(\mu\times \mu)$ stands for the product measure and $\mathscr{F}\otimes \mathscr{F}$ for the product $\sigma$-algebra]