Field And sigma field understanding
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In field, the condition that differentiates field from sigma field is if A1,A2....ϵ field, then $bigcuplimits_{i=1}^{n} $$A{i}$ must ϵ field,but if we take limit n tends to infinity, then it essentially becomes the condition for sigma field? by this logic field and sigma field are the same. What am i understanding wrong?
probability measure-theory field-theory
asked Nov 21 at 22:44
Yashasvi Grover
1092
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In field, the condition that differentiates field from sigma field is if A1,A2....ϵ field, then $bigcuplimits_{i=1}^{n} $$A{i}$ must ϵ field,but if we take limit n tends to infinity, then it essentially becomes the condition for sigma field? by this logic field and sigma field are the same. What am i understanding wrong?
probability measure-theory field-theory
asked Nov 21 at 22:44
Yashasvi Grover
1092
Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In field, the condition that differentiates field from sigma field is if A1,A2....ϵ field, then $bigcuplimits_{i=1}^{n} $$A{i}$ must ϵ field,but if we take limit n tends to infinity, then it essentially becomes the condition for sigma field? by this logic field and sigma field are the same. What am i understanding wrong?
probability measure-theory field-theory
asked Nov 21 at 22:44
Yashasvi Grover
1092
In field, the condition that differentiates field from sigma field is if A1,A2....ϵ field, then $bigcuplimits_{i=1}^{n} $$A{i}$ must ϵ field,but if we take limit n tends to infinity, then it essentially becomes the condition for sigma field? by this logic field and sigma field are the same. What am i understanding wrong?
probability measure-theory field-theory
probability measure-theory field-theory
asked Nov 21 at 22:44
Yashasvi Grover
1092
asked Nov 21 at 22:44
Yashasvi Grover
1092
edited Nov 21 at 22:52
asked Nov 21 at 22:44
Yashasvi Grover
1092
asked Nov 21 at 22:44
Yashasvi Grover
1092
asked Nov 21 at 22:44
Yashasvi Grover
1092
1092
Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42
|
show 1 more comment
Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42
Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42
|
show 1 more comment
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Sorry but your question is really unclear. $sigma -$field are stable by countable union, whereas field are not... so it's not the same.
– idm
Nov 21 at 22:59
@idm that part is where i am confused. Sigma field allows countable unions which are infinite.Field has finite unions which can be uncountable.(am i right?) Now my understanding is a field which has countable unions will always be a sigma field as by taking limit n to infinity we can extend to be sigma field. Am i right? can there be a field which has countable unions but is not a sigma field? Also can you please give an example of field which is not a sigma field?
– Yashasvi Grover
Nov 21 at 23:08
@idm "σ− field are stable by countable union, whereas field are not.." Please what do you mean by this then. Can you elaborate a bit.Clearly i have some basic conceptual flaw.
– Yashasvi Grover
Nov 21 at 23:15
Take for example the set ${Ssubset mathbb Rmid S text{open or close}}$. It's a field, but not a $sigma -$field since for example $bigcap_{ninmathbb N^*}[0,1+1/n]=[0,1)$ is neither open not closed.
– idm
Nov 21 at 23:26
@idm please point out the flaw now. It is a field because ⋂n∈N∗[0,1+1/n] stops at n=k for some finite k so it is closed and hence belongs to field while if we take n= infinity, it becomes open on only one side so it does not belong to the set hence it is not sigma field. But my question is why we stop at n=k, it is still closed for n=k+1. So n=k and n=k+1 intersection must also lie in field by definition and using this recursively, again we conclude that limit n tends to infinity must also lie it to be a field which is not true.Where am i wrong here?
– Yashasvi Grover
Nov 21 at 23:42