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Is this method of definition of Borel $sigma$-fields complete?
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A Borel $sigma$-field is the smallest $sigma$ field that contains $(-infty,x],: x in mathbbR$
Is this definition complete? I have been trying to use unions, intersections and complements of such sets to prove that open sets (say, $(a,b)$) belong to the Borel $sigma$-field, but I have been unable to do so. I don't understand how to prove open intervals belong in the field. I have only been able to prove that $(a,b]$ belong in the field. I did this by observing that $(-infty,a]$ belongs, so its complement, i.e. $(a, infty)$ belongs. So intersection of $(a,infty)$ and $(-infty, b]$, i.e. $(a, b]$ belongs. But I am stuck here.
Is the definition complete? Or do we need to assume that intervals of type $[x, infty)$ also belong? If the definition is complete, how do we prove that the Borel $sigma$-field includes open intervals?
probability-theory measure-theory borel-sets borel-measures
asked Jul 28 at 14:03
FreezingFire
696220
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up vote
3
down vote
favorite
A Borel $sigma$-field is the smallest $sigma$ field that contains $(-infty,x],: x in mathbbR$
Is this definition complete? I have been trying to use unions, intersections and complements of such sets to prove that open sets (say, $(a,b)$) belong to the Borel $sigma$-field, but I have been unable to do so. I don't understand how to prove open intervals belong in the field. I have only been able to prove that $(a,b]$ belong in the field. I did this by observing that $(-infty,a]$ belongs, so its complement, i.e. $(a, infty)$ belongs. So intersection of $(a,infty)$ and $(-infty, b]$, i.e. $(a, b]$ belongs. But I am stuck here.
Is the definition complete? Or do we need to assume that intervals of type $[x, infty)$ also belong? If the definition is complete, how do we prove that the Borel $sigma$-field includes open intervals?
probability-theory measure-theory borel-sets borel-measures
asked Jul 28 at 14:03
FreezingFire
696220
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
A Borel $sigma$-field is the smallest $sigma$ field that contains $(-infty,x],: x in mathbbR$
Is this definition complete? I have been trying to use unions, intersections and complements of such sets to prove that open sets (say, $(a,b)$) belong to the Borel $sigma$-field, but I have been unable to do so. I don't understand how to prove open intervals belong in the field. I have only been able to prove that $(a,b]$ belong in the field. I did this by observing that $(-infty,a]$ belongs, so its complement, i.e. $(a, infty)$ belongs. So intersection of $(a,infty)$ and $(-infty, b]$, i.e. $(a, b]$ belongs. But I am stuck here.
Is the definition complete? Or do we need to assume that intervals of type $[x, infty)$ also belong? If the definition is complete, how do we prove that the Borel $sigma$-field includes open intervals?
probability-theory measure-theory borel-sets borel-measures
asked Jul 28 at 14:03
FreezingFire
696220
A Borel $sigma$-field is the smallest $sigma$ field that contains $(-infty,x],: x in mathbbR$
Is this definition complete? I have been trying to use unions, intersections and complements of such sets to prove that open sets (say, $(a,b)$) belong to the Borel $sigma$-field, but I have been unable to do so. I don't understand how to prove open intervals belong in the field. I have only been able to prove that $(a,b]$ belong in the field. I did this by observing that $(-infty,a]$ belongs, so its complement, i.e. $(a, infty)$ belongs. So intersection of $(a,infty)$ and $(-infty, b]$, i.e. $(a, b]$ belongs. But I am stuck here.
Is the definition complete? Or do we need to assume that intervals of type $[x, infty)$ also belong? If the definition is complete, how do we prove that the Borel $sigma$-field includes open intervals?
probability-theory measure-theory borel-sets borel-measures
asked Jul 28 at 14:03
FreezingFire
696220
asked Jul 28 at 14:03
FreezingFire
696220
asked Jul 28 at 14:03
FreezingFire
696220
asked Jul 28 at 14:03
FreezingFire
696220
696220
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Let $newcommandcfmathcalPcf$ be the sigma-field of Borel sets.
You can prove
$(-infty,b)incf$ by observing $(-infty,b)=bigcup_n=1^infty(-infty,b-1/n]$. You know $(a,infty)incf$, so $(a,b)=(a,infty)
cap(-infty,b)incf$ etc.
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Let $newcommandcfmathcalPcf$ be the sigma-field of Borel sets.
You can prove
$(-infty,b)incf$ by observing $(-infty,b)=bigcup_n=1^infty(-infty,b-1/n]$. You know $(a,infty)incf$, so $(a,b)=(a,infty)
cap(-infty,b)incf$ etc.
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
add a comment |Â
up vote
1
down vote
accepted
Let $newcommandcfmathcalPcf$ be the sigma-field of Borel sets.
You can prove
$(-infty,b)incf$ by observing $(-infty,b)=bigcup_n=1^infty(-infty,b-1/n]$. You know $(a,infty)incf$, so $(a,b)=(a,infty)
cap(-infty,b)incf$ etc.
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $newcommandcfmathcalPcf$ be the sigma-field of Borel sets.
You can prove
$(-infty,b)incf$ by observing $(-infty,b)=bigcup_n=1^infty(-infty,b-1/n]$. You know $(a,infty)incf$, so $(a,b)=(a,infty)
cap(-infty,b)incf$ etc.
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
Let $newcommandcfmathcalPcf$ be the sigma-field of Borel sets.
You can prove
$(-infty,b)incf$ by observing $(-infty,b)=bigcup_n=1^infty(-infty,b-1/n]$. You know $(a,infty)incf$, so $(a,b)=(a,infty)
cap(-infty,b)incf$ etc.
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
answered Jul 28 at 14:08
Lord Shark the Unknown
84.6k950111
84.6k950111
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
add a comment |Â
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
Nice answer! Also I noticed that $(-infty,b)$ is a countably infinite union of sets already known to be in the field, which is why this approach works. And I loved how this approach expressed an open set as a limit of closed sets (closed at one end). Thank you for your answer!
– FreezingFire
Jul 28 at 14:20
add a comment |Â
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