collection of sigma algebra
How to solve this exercise?
If $A$ is a collection of $sigma$-algebra. Show that $bigcap A=bigcap_{Fin A}F$ is a $sigma$-algebra.
real-analysis measure-theory
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
|
show 1 more comment
How to solve this exercise?
If $A$ is a collection of $sigma$-algebra. Show that $bigcap A=bigcap_{Fin A}F$ is a $sigma$-algebra.
real-analysis measure-theory
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49
|
show 1 more comment
How to solve this exercise?
If $A$ is a collection of $sigma$-algebra. Show that $bigcap A=bigcap_{Fin A}F$ is a $sigma$-algebra.
real-analysis measure-theory
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
How to solve this exercise?
If $A$ is a collection of $sigma$-algebra. Show that $bigcap A=bigcap_{Fin A}F$ is a $sigma$-algebra.
real-analysis measure-theory
real-analysis measure-theory
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
edited Nov 18 '18 at 18:39
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
asked Nov 18 '18 at 18:35
Nathan Explosion
1476
1476
Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49
|
show 1 more comment
Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49
Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49
|
show 1 more comment
1 Answer
1
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votes
It is clear that $emptyset in bigcap A$ (?).
Given $Ein bigcap A$, $Ein F$ for all $Fin A$ hence $E^c$ is in each $F$, i.e., $E^c in bigcap A$.
Given a sequence $(E_j) in bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $bigcup E_j in F ; forall Fin A$, that's equivalent to say that $bigcup E_i in bigcap A$...
So $bigcap A$ is closed under complement and countable union.
answered Nov 18 '18 at 22:10
Robson
771221
add a comment |
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1 Answer
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1 Answer
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It is clear that $emptyset in bigcap A$ (?).
Given $Ein bigcap A$, $Ein F$ for all $Fin A$ hence $E^c$ is in each $F$, i.e., $E^c in bigcap A$.
Given a sequence $(E_j) in bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $bigcup E_j in F ; forall Fin A$, that's equivalent to say that $bigcup E_i in bigcap A$...
So $bigcap A$ is closed under complement and countable union.
answered Nov 18 '18 at 22:10
Robson
771221
add a comment |
It is clear that $emptyset in bigcap A$ (?).
Given $Ein bigcap A$, $Ein F$ for all $Fin A$ hence $E^c$ is in each $F$, i.e., $E^c in bigcap A$.
Given a sequence $(E_j) in bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $bigcup E_j in F ; forall Fin A$, that's equivalent to say that $bigcup E_i in bigcap A$...
So $bigcap A$ is closed under complement and countable union.
answered Nov 18 '18 at 22:10
Robson
771221
add a comment |
It is clear that $emptyset in bigcap A$ (?).
Given $Ein bigcap A$, $Ein F$ for all $Fin A$ hence $E^c$ is in each $F$, i.e., $E^c in bigcap A$.
Given a sequence $(E_j) in bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $bigcup E_j in F ; forall Fin A$, that's equivalent to say that $bigcup E_i in bigcap A$...
So $bigcap A$ is closed under complement and countable union.
answered Nov 18 '18 at 22:10
Robson
771221
It is clear that $emptyset in bigcap A$ (?).
Given $Ein bigcap A$, $Ein F$ for all $Fin A$ hence $E^c$ is in each $F$, i.e., $E^c in bigcap A$.
Given a sequence $(E_j) in bigcap A$, each $E_j$ is in every $F$, so the sequence is contained in all $F$ and hence $bigcup E_j in F ; forall Fin A$, that's equivalent to say that $bigcup E_i in bigcap A$...
So $bigcap A$ is closed under complement and countable union.
answered Nov 18 '18 at 22:10
Robson
771221
edited Nov 25 '18 at 22:45
answered Nov 18 '18 at 22:10
Robson
771221
answered Nov 18 '18 at 22:10
Robson
771221
answered Nov 18 '18 at 22:10
Robson
771221
771221
add a comment |
add a comment |
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Do you mean $bigcap A=bigcap_{Fin A}F$?
– ervx
Nov 18 '18 at 18:38
@ervx yes, sorry for my typo..
– Nathan Explosion
Nov 18 '18 at 18:40
What do you know about $sigma$-algebras? What properties to we need to check to see whether or not $bigcap A$ is a $sigma$-algebra?
– ervx
Nov 18 '18 at 18:41
@ervx I guess to prove $bigcap A$ is closed under complement?
– Nathan Explosion
Nov 18 '18 at 18:43
So if you take a set $Bincap A$, then it must be that $Bin F$ for each $F$ in $A$. But each such $F$ is a $sigma$-algebra, so $B^{c}in F$ for each such $F$. Thus, $B^{c}in cap A$. Does that make sense? This is part of what you need to show. What else?
– ervx
Nov 18 '18 at 18:49