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What is the right concept for completion of a sub-sigma algebra?
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Let $E$ be a set, $mathcalE$ a $sigma$-algebra on $E$ and $mu : mathcalE to [0, infty]$ a measure on $mathcalE$.
Set $mathcalN^mu := A subseteq E mid exists B in mathcalE, A subseteq B, mu(B) = 0 $ as the collection of subsets of $mu$-null sets. Then $mathcalE^mu := sigma(mathcalE cup mathcalN^mu)$ is the completion of $mathcalE$: it is the smallest $mu$-complete $sigma$-algebra larger than $mathcalE$.
Let $mathcalF subseteq mathcalE$ be a sub-$sigma$-algebra. Then the restriction $mu|_mathcalF : mathcalF to [0, infty]$ is a measure on $mathcalF$. Hence we can consider the $mu|_mathcalF$-completion $mathcalF^mu = sigma(mathcalF cup mathcalN^mu)$ of $mathcalF$.
I often see in books on stochastic processes also the following definition: $mathcalF^mu := sigma(mathcalF cup mathcalN^mu)$.
The difference is that in $mathcalF^mu$ we add in all (subsets of) $mu$-null sets of $mathcalE$ (and not only those of $mathcalF$).
It holds: $mathcalF^mu subseteq mathcalF^mu subseteq mathcalE^mu$ and all the inclusions may be strict. For an extreme case, just consider $mathcalF$ the trivial $sigma$-algebra.
What is the canonical or natural definition of a $mu$-completion of a sub-$sigma$-algebra - especially in the context of stochastic processes and completions of filtrations? (It would also be interesting to have an aswer in categorical terms, explaining the naturality of the construction.)
probability-theory measure-theory stochastic-processes category-theory
asked Jul 20 at 14:02
yadaddy
1,209815
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up vote
2
down vote
favorite
Let $E$ be a set, $mathcalE$ a $sigma$-algebra on $E$ and $mu : mathcalE to [0, infty]$ a measure on $mathcalE$.
Set $mathcalN^mu := A subseteq E mid exists B in mathcalE, A subseteq B, mu(B) = 0 $ as the collection of subsets of $mu$-null sets. Then $mathcalE^mu := sigma(mathcalE cup mathcalN^mu)$ is the completion of $mathcalE$: it is the smallest $mu$-complete $sigma$-algebra larger than $mathcalE$.
Let $mathcalF subseteq mathcalE$ be a sub-$sigma$-algebra. Then the restriction $mu|_mathcalF : mathcalF to [0, infty]$ is a measure on $mathcalF$. Hence we can consider the $mu|_mathcalF$-completion $mathcalF^mu = sigma(mathcalF cup mathcalN^mu)$ of $mathcalF$.
I often see in books on stochastic processes also the following definition: $mathcalF^mu := sigma(mathcalF cup mathcalN^mu)$.
The difference is that in $mathcalF^mu$ we add in all (subsets of) $mu$-null sets of $mathcalE$ (and not only those of $mathcalF$).
It holds: $mathcalF^mu subseteq mathcalF^mu subseteq mathcalE^mu$ and all the inclusions may be strict. For an extreme case, just consider $mathcalF$ the trivial $sigma$-algebra.
What is the canonical or natural definition of a $mu$-completion of a sub-$sigma$-algebra - especially in the context of stochastic processes and completions of filtrations? (It would also be interesting to have an aswer in categorical terms, explaining the naturality of the construction.)
probability-theory measure-theory stochastic-processes category-theory
asked Jul 20 at 14:02
yadaddy
1,209815
2
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $E$ be a set, $mathcalE$ a $sigma$-algebra on $E$ and $mu : mathcalE to [0, infty]$ a measure on $mathcalE$.
Set $mathcalN^mu := A subseteq E mid exists B in mathcalE, A subseteq B, mu(B) = 0 $ as the collection of subsets of $mu$-null sets. Then $mathcalE^mu := sigma(mathcalE cup mathcalN^mu)$ is the completion of $mathcalE$: it is the smallest $mu$-complete $sigma$-algebra larger than $mathcalE$.
Let $mathcalF subseteq mathcalE$ be a sub-$sigma$-algebra. Then the restriction $mu|_mathcalF : mathcalF to [0, infty]$ is a measure on $mathcalF$. Hence we can consider the $mu|_mathcalF$-completion $mathcalF^mu = sigma(mathcalF cup mathcalN^mu)$ of $mathcalF$.
I often see in books on stochastic processes also the following definition: $mathcalF^mu := sigma(mathcalF cup mathcalN^mu)$.
The difference is that in $mathcalF^mu$ we add in all (subsets of) $mu$-null sets of $mathcalE$ (and not only those of $mathcalF$).
It holds: $mathcalF^mu subseteq mathcalF^mu subseteq mathcalE^mu$ and all the inclusions may be strict. For an extreme case, just consider $mathcalF$ the trivial $sigma$-algebra.
What is the canonical or natural definition of a $mu$-completion of a sub-$sigma$-algebra - especially in the context of stochastic processes and completions of filtrations? (It would also be interesting to have an aswer in categorical terms, explaining the naturality of the construction.)
probability-theory measure-theory stochastic-processes category-theory
asked Jul 20 at 14:02
yadaddy
1,209815
Let $E$ be a set, $mathcalE$ a $sigma$-algebra on $E$ and $mu : mathcalE to [0, infty]$ a measure on $mathcalE$.
Set $mathcalN^mu := A subseteq E mid exists B in mathcalE, A subseteq B, mu(B) = 0 $ as the collection of subsets of $mu$-null sets. Then $mathcalE^mu := sigma(mathcalE cup mathcalN^mu)$ is the completion of $mathcalE$: it is the smallest $mu$-complete $sigma$-algebra larger than $mathcalE$.
Let $mathcalF subseteq mathcalE$ be a sub-$sigma$-algebra. Then the restriction $mu|_mathcalF : mathcalF to [0, infty]$ is a measure on $mathcalF$. Hence we can consider the $mu|_mathcalF$-completion $mathcalF^mu = sigma(mathcalF cup mathcalN^mu)$ of $mathcalF$.
I often see in books on stochastic processes also the following definition: $mathcalF^mu := sigma(mathcalF cup mathcalN^mu)$.
The difference is that in $mathcalF^mu$ we add in all (subsets of) $mu$-null sets of $mathcalE$ (and not only those of $mathcalF$).
It holds: $mathcalF^mu subseteq mathcalF^mu subseteq mathcalE^mu$ and all the inclusions may be strict. For an extreme case, just consider $mathcalF$ the trivial $sigma$-algebra.
What is the canonical or natural definition of a $mu$-completion of a sub-$sigma$-algebra - especially in the context of stochastic processes and completions of filtrations? (It would also be interesting to have an aswer in categorical terms, explaining the naturality of the construction.)
probability-theory measure-theory stochastic-processes category-theory
asked Jul 20 at 14:02
yadaddy
1,209815
asked Jul 20 at 14:02
yadaddy
1,209815
asked Jul 20 at 14:02
yadaddy
1,209815
asked Jul 20 at 14:02
yadaddy
1,209815
1,209815
2
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40
add a comment |Â
2
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40
2
2
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40
add a comment |Â
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2
I think, we most often want to take the $sigma$-algebra of measurable sets as big as possible. That aligns with the second definition of completion of $mathcal F$, when an ambient $sigma$-algebra $mathcal E$ with an ambient measure is in our hands.
– Berci
Jul 21 at 23:40