A family of sets that is not a $sigma$-algebra
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Let $X$ be a not empty set.
I must prove that the family $mathcal{A}$ defined as follows
$$mathcal{A}=bigg{Ainmathcal{P}(X);|;A;text{is finite or $A^c$ is finite}bigg}$$ it is not a $sigma$-algebra if $X$ is infinite.
Obviously $mathcal{A}$ is an algebra. To show that it is not a $sigma$-algebra I reasoned as follows: fixed a sequence ${x_n}_{ninmathbb{N}}$ of elements of $X$ such that $x_ine x_j$ for $ine j$ and let $A_n$ be the set consisting of the single point $x_n$, in symbols $A_n={x_n}$ for all $ninmathbb{N}$. At this point I consider the sets $A_2,A_4dots ,A_{2k},dots$ where $kinmathbb{N}$; I observe that $A_{2k}inmathcal{A}$ for all $kinmathbb{N}$, but the union $$bigcup_{k=1}^{+infty}A_{2k}=bigg{x_2,x_4,dots,x_{2k},dotsbigg}$$
is infinite. On the other hand
begin{equation}
begin{split}
bigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^c=Xsetminusbigcup_{k=1}^{+infty}A_{2k}=& Xcapbigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^{c}=Xcapbigg[bigcup_{k=1}^{+infty}A_{2k+1}bigg]\
=&bigcup_{k=1}^{+infty}A_{2k+1}cuptilde{A},
end{split}
end{equation}
where $tilde{A}$ it is what remains if $X$ is not countable, if $X$ is countable $tilde{A}=emptyset$. Therefore $cup_k A_{2k}$ and $[cup_k A_{2k}]^c$ are both infinite, then $$bigcup_{n=1}^{+infty} A_nnotinmathcal{A}.$$
it's correct? Thanks!
proof-verification proof-writing proof-explanation
add a comment |
up vote
1
down vote
favorite
Let $X$ be a not empty set.
I must prove that the family $mathcal{A}$ defined as follows
$$mathcal{A}=bigg{Ainmathcal{P}(X);|;A;text{is finite or $A^c$ is finite}bigg}$$ it is not a $sigma$-algebra if $X$ is infinite.
Obviously $mathcal{A}$ is an algebra. To show that it is not a $sigma$-algebra I reasoned as follows: fixed a sequence ${x_n}_{ninmathbb{N}}$ of elements of $X$ such that $x_ine x_j$ for $ine j$ and let $A_n$ be the set consisting of the single point $x_n$, in symbols $A_n={x_n}$ for all $ninmathbb{N}$. At this point I consider the sets $A_2,A_4dots ,A_{2k},dots$ where $kinmathbb{N}$; I observe that $A_{2k}inmathcal{A}$ for all $kinmathbb{N}$, but the union $$bigcup_{k=1}^{+infty}A_{2k}=bigg{x_2,x_4,dots,x_{2k},dotsbigg}$$
is infinite. On the other hand
begin{equation}
begin{split}
bigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^c=Xsetminusbigcup_{k=1}^{+infty}A_{2k}=& Xcapbigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^{c}=Xcapbigg[bigcup_{k=1}^{+infty}A_{2k+1}bigg]\
=&bigcup_{k=1}^{+infty}A_{2k+1}cuptilde{A},
end{split}
end{equation}
where $tilde{A}$ it is what remains if $X$ is not countable, if $X$ is countable $tilde{A}=emptyset$. Therefore $cup_k A_{2k}$ and $[cup_k A_{2k}]^c$ are both infinite, then $$bigcup_{n=1}^{+infty} A_nnotinmathcal{A}.$$
it's correct? Thanks!
proof-verification proof-writing proof-explanation
Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $X$ be a not empty set.
I must prove that the family $mathcal{A}$ defined as follows
$$mathcal{A}=bigg{Ainmathcal{P}(X);|;A;text{is finite or $A^c$ is finite}bigg}$$ it is not a $sigma$-algebra if $X$ is infinite.
Obviously $mathcal{A}$ is an algebra. To show that it is not a $sigma$-algebra I reasoned as follows: fixed a sequence ${x_n}_{ninmathbb{N}}$ of elements of $X$ such that $x_ine x_j$ for $ine j$ and let $A_n$ be the set consisting of the single point $x_n$, in symbols $A_n={x_n}$ for all $ninmathbb{N}$. At this point I consider the sets $A_2,A_4dots ,A_{2k},dots$ where $kinmathbb{N}$; I observe that $A_{2k}inmathcal{A}$ for all $kinmathbb{N}$, but the union $$bigcup_{k=1}^{+infty}A_{2k}=bigg{x_2,x_4,dots,x_{2k},dotsbigg}$$
is infinite. On the other hand
begin{equation}
begin{split}
bigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^c=Xsetminusbigcup_{k=1}^{+infty}A_{2k}=& Xcapbigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^{c}=Xcapbigg[bigcup_{k=1}^{+infty}A_{2k+1}bigg]\
=&bigcup_{k=1}^{+infty}A_{2k+1}cuptilde{A},
end{split}
end{equation}
where $tilde{A}$ it is what remains if $X$ is not countable, if $X$ is countable $tilde{A}=emptyset$. Therefore $cup_k A_{2k}$ and $[cup_k A_{2k}]^c$ are both infinite, then $$bigcup_{n=1}^{+infty} A_nnotinmathcal{A}.$$
it's correct? Thanks!
proof-verification proof-writing proof-explanation
Let $X$ be a not empty set.
I must prove that the family $mathcal{A}$ defined as follows
$$mathcal{A}=bigg{Ainmathcal{P}(X);|;A;text{is finite or $A^c$ is finite}bigg}$$ it is not a $sigma$-algebra if $X$ is infinite.
Obviously $mathcal{A}$ is an algebra. To show that it is not a $sigma$-algebra I reasoned as follows: fixed a sequence ${x_n}_{ninmathbb{N}}$ of elements of $X$ such that $x_ine x_j$ for $ine j$ and let $A_n$ be the set consisting of the single point $x_n$, in symbols $A_n={x_n}$ for all $ninmathbb{N}$. At this point I consider the sets $A_2,A_4dots ,A_{2k},dots$ where $kinmathbb{N}$; I observe that $A_{2k}inmathcal{A}$ for all $kinmathbb{N}$, but the union $$bigcup_{k=1}^{+infty}A_{2k}=bigg{x_2,x_4,dots,x_{2k},dotsbigg}$$
is infinite. On the other hand
begin{equation}
begin{split}
bigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^c=Xsetminusbigcup_{k=1}^{+infty}A_{2k}=& Xcapbigg[bigcup_{k=1}^{+infty}A_{2k}bigg]^{c}=Xcapbigg[bigcup_{k=1}^{+infty}A_{2k+1}bigg]\
=&bigcup_{k=1}^{+infty}A_{2k+1}cuptilde{A},
end{split}
end{equation}
where $tilde{A}$ it is what remains if $X$ is not countable, if $X$ is countable $tilde{A}=emptyset$. Therefore $cup_k A_{2k}$ and $[cup_k A_{2k}]^c$ are both infinite, then $$bigcup_{n=1}^{+infty} A_nnotinmathcal{A}.$$
it's correct? Thanks!
proof-verification proof-writing proof-explanation
proof-verification proof-writing proof-explanation
asked Nov 18 at 13:48
Jack J.
4441317
4441317
Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52
add a comment |
Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52
Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Your proof is correct.
A bit shorter:
Let $X$ be infinite.
Then we can write it as $X=Ycup Z$ where:
$Y$ and $Z$ are disjoint.
$Y$ is countably infinite.
$Z$ is infinite.
Then $Ynotinmathcal A$, but also $Y=bigcup_{yin Y}{y}$ where the RHS is a countable union of sets that are elements of $mathcal A$.
So $mathcal A$ is not closed under countable unions, hence is not a $sigma$-algebra.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Your proof is correct.
A bit shorter:
Let $X$ be infinite.
Then we can write it as $X=Ycup Z$ where:
$Y$ and $Z$ are disjoint.
$Y$ is countably infinite.
$Z$ is infinite.
Then $Ynotinmathcal A$, but also $Y=bigcup_{yin Y}{y}$ where the RHS is a countable union of sets that are elements of $mathcal A$.
So $mathcal A$ is not closed under countable unions, hence is not a $sigma$-algebra.
add a comment |
up vote
2
down vote
accepted
Your proof is correct.
A bit shorter:
Let $X$ be infinite.
Then we can write it as $X=Ycup Z$ where:
$Y$ and $Z$ are disjoint.
$Y$ is countably infinite.
$Z$ is infinite.
Then $Ynotinmathcal A$, but also $Y=bigcup_{yin Y}{y}$ where the RHS is a countable union of sets that are elements of $mathcal A$.
So $mathcal A$ is not closed under countable unions, hence is not a $sigma$-algebra.
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Your proof is correct.
A bit shorter:
Let $X$ be infinite.
Then we can write it as $X=Ycup Z$ where:
$Y$ and $Z$ are disjoint.
$Y$ is countably infinite.
$Z$ is infinite.
Then $Ynotinmathcal A$, but also $Y=bigcup_{yin Y}{y}$ where the RHS is a countable union of sets that are elements of $mathcal A$.
So $mathcal A$ is not closed under countable unions, hence is not a $sigma$-algebra.
Your proof is correct.
A bit shorter:
Let $X$ be infinite.
Then we can write it as $X=Ycup Z$ where:
$Y$ and $Z$ are disjoint.
$Y$ is countably infinite.
$Z$ is infinite.
Then $Ynotinmathcal A$, but also $Y=bigcup_{yin Y}{y}$ where the RHS is a countable union of sets that are elements of $mathcal A$.
So $mathcal A$ is not closed under countable unions, hence is not a $sigma$-algebra.
edited Nov 18 at 20:37
answered Nov 18 at 13:59
drhab
94.9k543125
94.9k543125
add a comment |
add a comment |
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Your solution is perfectly fine, you are using that a countable collection of sets in a sigma algebra stays within the sigma algebra, which is part of the definition of a sigma algebra and you are right in proving that this doesn’t hold here
– Sorin Tirc
Nov 18 at 13:52
Your argument is essentially correct. You struggle a bit more than necessary showing the complement of the union you found is infinite.
– Ethan Bolker
Nov 18 at 13:52