Question about the collection of all cylindrical subsets of continuous function on $[0,1]$ that vanish at $0$
$begingroup$
Let $C$ be the set of all continuous real-valued functions on $[0,1]$ with the property if $fin C$ then $f(0)=0$. My book defines a cylindrical subset $A$ of $C$ as $A={fin C : (f(t_1),...,f(t_n) in U}$, where $0<t_1<...<t_n le 1$ and $U in mathscr B(mathbb R^n)$ where $mathscr B(mathbb R^n)$ is the Borel sigma-field of $mathbb R^n$. Now let $mathscr R$ be the collection of all cylindrical subsets of $C$. My book claims that $mathscr R$ is a field but not a sigma-field.
My question is what is $mathscr R$? Do we fix $n$ and $(t_1,...,t_n)$ and say $mathscr R$ is ${A_U}_{Uin mathscr B(mathbb R^n)}$.
If not, say $A_1={f in C : (f(t_1),f(t_2)) in U}$ and $A_2={g in C : g(p_1) in V$ } where $U in mathscr B(mathbb R^2)$ and $Vin mathscr B(mathbb R)$ then what is $A_1 cup A_2$ or $A_1 cap A_2$ and why is this going to be in $mathscr R$?
measure-theory
asked Jan 14 at 20:55
alpastalpast
468314
$endgroup$
add a comment |
$begingroup$
Let $C$ be the set of all continuous real-valued functions on $[0,1]$ with the property if $fin C$ then $f(0)=0$. My book defines a cylindrical subset $A$ of $C$ as $A={fin C : (f(t_1),...,f(t_n) in U}$, where $0<t_1<...<t_n le 1$ and $U in mathscr B(mathbb R^n)$ where $mathscr B(mathbb R^n)$ is the Borel sigma-field of $mathbb R^n$. Now let $mathscr R$ be the collection of all cylindrical subsets of $C$. My book claims that $mathscr R$ is a field but not a sigma-field.
My question is what is $mathscr R$? Do we fix $n$ and $(t_1,...,t_n)$ and say $mathscr R$ is ${A_U}_{Uin mathscr B(mathbb R^n)}$.
If not, say $A_1={f in C : (f(t_1),f(t_2)) in U}$ and $A_2={g in C : g(p_1) in V$ } where $U in mathscr B(mathbb R^2)$ and $Vin mathscr B(mathbb R)$ then what is $A_1 cup A_2$ or $A_1 cap A_2$ and why is this going to be in $mathscr R$?
measure-theory
asked Jan 14 at 20:55
alpastalpast
468314
$endgroup$
add a comment |
$begingroup$
Let $C$ be the set of all continuous real-valued functions on $[0,1]$ with the property if $fin C$ then $f(0)=0$. My book defines a cylindrical subset $A$ of $C$ as $A={fin C : (f(t_1),...,f(t_n) in U}$, where $0<t_1<...<t_n le 1$ and $U in mathscr B(mathbb R^n)$ where $mathscr B(mathbb R^n)$ is the Borel sigma-field of $mathbb R^n$. Now let $mathscr R$ be the collection of all cylindrical subsets of $C$. My book claims that $mathscr R$ is a field but not a sigma-field.
My question is what is $mathscr R$? Do we fix $n$ and $(t_1,...,t_n)$ and say $mathscr R$ is ${A_U}_{Uin mathscr B(mathbb R^n)}$.
If not, say $A_1={f in C : (f(t_1),f(t_2)) in U}$ and $A_2={g in C : g(p_1) in V$ } where $U in mathscr B(mathbb R^2)$ and $Vin mathscr B(mathbb R)$ then what is $A_1 cup A_2$ or $A_1 cap A_2$ and why is this going to be in $mathscr R$?
measure-theory
asked Jan 14 at 20:55
alpastalpast
468314
$endgroup$
Let $C$ be the set of all continuous real-valued functions on $[0,1]$ with the property if $fin C$ then $f(0)=0$. My book defines a cylindrical subset $A$ of $C$ as $A={fin C : (f(t_1),...,f(t_n) in U}$, where $0<t_1<...<t_n le 1$ and $U in mathscr B(mathbb R^n)$ where $mathscr B(mathbb R^n)$ is the Borel sigma-field of $mathbb R^n$. Now let $mathscr R$ be the collection of all cylindrical subsets of $C$. My book claims that $mathscr R$ is a field but not a sigma-field.
My question is what is $mathscr R$? Do we fix $n$ and $(t_1,...,t_n)$ and say $mathscr R$ is ${A_U}_{Uin mathscr B(mathbb R^n)}$.
If not, say $A_1={f in C : (f(t_1),f(t_2)) in U}$ and $A_2={g in C : g(p_1) in V$ } where $U in mathscr B(mathbb R^2)$ and $Vin mathscr B(mathbb R)$ then what is $A_1 cup A_2$ or $A_1 cap A_2$ and why is this going to be in $mathscr R$?
measure-theory
measure-theory
asked Jan 14 at 20:55
alpastalpast
468314
asked Jan 14 at 20:55
alpastalpast
468314
asked Jan 14 at 20:55
alpastalpast
468314
asked Jan 14 at 20:55
alpastalpast
468314
asked Jan 14 at 20:55
alpastalpast
468314
468314
add a comment |
add a comment |
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