The consequence of $mu$-measurability from Bogachev's book
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I am reading Bogachev's book "Measure Theory" which is in my opinion is very good book on measure theory. Let me ask you the following question:
Definition: Let $mu$ - non-negative set function with domain $mathcal{A}subset 2^X$. The set $A$ is called $mu$-measurable if
for any $varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$
such that $mu^*(Atriangle A_{varepsilon})<varepsilon,$ where
$mu^*$-outer measure.
The set of all $mu$-measurable sets is denoted by
$mathcal{A}_{mu}$.
One thing began to confuse me during reading. When author takes some set $Ain mathcal{A}_{mu}$ and then he considers $mu(A)$. This confuses me a lot.
We know that $mathcal{A}subset mathcal{A}_{mu}$ and when $Ain mathcal{A}_{mu}$ it means that $A$ is $mu$-measurable set. But $mu$ is defined only on $mathcal{A}$. Why he does not consider $mu^*(A)$?
More precisely, if $A$ is $mu$-measurable ($Ain mathcal{A}_{mu}$) then why $mu(A)$ makes sense? and does $mu(A)=mu^*(A)$?
P.S. I know the fact that $mathcal{A}_{mu}$ is $sigma$-algebra and $mu$ can be uniquely extended from algebra $mathcal{A}$ to $sigma$-algebra $mathcal{A}_{mu}$. And this extension is given by $mu^*$.
I would be very grateful if somebody can detailed answer. Because it is important two understand such things.
EDIT: I will divide my questions into 3 parts.
1) Corollary 1.5.8. (page 21).
I was not able to understand this part. He just uses the definition of outer measure $mu^*$ but not the definition of $mu$-measurability.
My approach: Since $A$ is $mu$-measurable then $forall varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$ such that $mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using monotonicity of $mu^*$ we have $mu^*(A-A_{varepsilon})leq mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using subadditivity of $mu^*$ we have $mu^*(A)-mu^*(A_{varepsilon})leq mu^*(A-A_{varepsilon})<varepsilon.$ But since $A_{varepsilon}in mathcal{A}$ then $mu^*(A_{varepsilon})=mu(A_{varepsilon})$ so $mu^*(A)-mu(A_{varepsilon})<varepsilon$. We have that $mu^*(A)$ is finite so we can apply the definition of outer measure. Is it correct? But I am sure that it is OK.
2)
He takes $Bin mathcal{A}_{mu}$ but why $mu(B)$ makes sense? $mu(B)$ is defined only on $malcal{A}$ but $B$ may not be in $mathcal{A}$. This also confusing me a lot.
3)
The same question $B$ and $C$ are $mu$-measurable sets but why $mu(B)$ and $mu(C)$ makes sense?
I guess that all these questions are related to each. So would be very grateful for detailed answer and help!
EDIT 2: Also one moment which I have forgot to ask in the previous questions:
In the above proof note two moments which I have underlined with red line. Why Bogachev writes $mu^*(A)=mu^*(A'')$ but in he writes just $mu(B)=mu^*(X-A)$. The first is OK and I am agree with that but in the second he omits $mu^*$ and just write $mu(B)$. Could you explain it, please?
measure-theory
add a comment |
up vote
0
down vote
favorite
I am reading Bogachev's book "Measure Theory" which is in my opinion is very good book on measure theory. Let me ask you the following question:
Definition: Let $mu$ - non-negative set function with domain $mathcal{A}subset 2^X$. The set $A$ is called $mu$-measurable if
for any $varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$
such that $mu^*(Atriangle A_{varepsilon})<varepsilon,$ where
$mu^*$-outer measure.
The set of all $mu$-measurable sets is denoted by
$mathcal{A}_{mu}$.
One thing began to confuse me during reading. When author takes some set $Ain mathcal{A}_{mu}$ and then he considers $mu(A)$. This confuses me a lot.
We know that $mathcal{A}subset mathcal{A}_{mu}$ and when $Ain mathcal{A}_{mu}$ it means that $A$ is $mu$-measurable set. But $mu$ is defined only on $mathcal{A}$. Why he does not consider $mu^*(A)$?
More precisely, if $A$ is $mu$-measurable ($Ain mathcal{A}_{mu}$) then why $mu(A)$ makes sense? and does $mu(A)=mu^*(A)$?
P.S. I know the fact that $mathcal{A}_{mu}$ is $sigma$-algebra and $mu$ can be uniquely extended from algebra $mathcal{A}$ to $sigma$-algebra $mathcal{A}_{mu}$. And this extension is given by $mu^*$.
I would be very grateful if somebody can detailed answer. Because it is important two understand such things.
EDIT: I will divide my questions into 3 parts.
1) Corollary 1.5.8. (page 21).
I was not able to understand this part. He just uses the definition of outer measure $mu^*$ but not the definition of $mu$-measurability.
My approach: Since $A$ is $mu$-measurable then $forall varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$ such that $mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using monotonicity of $mu^*$ we have $mu^*(A-A_{varepsilon})leq mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using subadditivity of $mu^*$ we have $mu^*(A)-mu^*(A_{varepsilon})leq mu^*(A-A_{varepsilon})<varepsilon.$ But since $A_{varepsilon}in mathcal{A}$ then $mu^*(A_{varepsilon})=mu(A_{varepsilon})$ so $mu^*(A)-mu(A_{varepsilon})<varepsilon$. We have that $mu^*(A)$ is finite so we can apply the definition of outer measure. Is it correct? But I am sure that it is OK.
2)
He takes $Bin mathcal{A}_{mu}$ but why $mu(B)$ makes sense? $mu(B)$ is defined only on $malcal{A}$ but $B$ may not be in $mathcal{A}$. This also confusing me a lot.
3)
The same question $B$ and $C$ are $mu$-measurable sets but why $mu(B)$ and $mu(C)$ makes sense?
I guess that all these questions are related to each. So would be very grateful for detailed answer and help!
EDIT 2: Also one moment which I have forgot to ask in the previous questions:
In the above proof note two moments which I have underlined with red line. Why Bogachev writes $mu^*(A)=mu^*(A'')$ but in he writes just $mu(B)=mu^*(X-A)$. The first is OK and I am agree with that but in the second he omits $mu^*$ and just write $mu(B)$. Could you explain it, please?
measure-theory
1
Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am reading Bogachev's book "Measure Theory" which is in my opinion is very good book on measure theory. Let me ask you the following question:
Definition: Let $mu$ - non-negative set function with domain $mathcal{A}subset 2^X$. The set $A$ is called $mu$-measurable if
for any $varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$
such that $mu^*(Atriangle A_{varepsilon})<varepsilon,$ where
$mu^*$-outer measure.
The set of all $mu$-measurable sets is denoted by
$mathcal{A}_{mu}$.
One thing began to confuse me during reading. When author takes some set $Ain mathcal{A}_{mu}$ and then he considers $mu(A)$. This confuses me a lot.
We know that $mathcal{A}subset mathcal{A}_{mu}$ and when $Ain mathcal{A}_{mu}$ it means that $A$ is $mu$-measurable set. But $mu$ is defined only on $mathcal{A}$. Why he does not consider $mu^*(A)$?
More precisely, if $A$ is $mu$-measurable ($Ain mathcal{A}_{mu}$) then why $mu(A)$ makes sense? and does $mu(A)=mu^*(A)$?
P.S. I know the fact that $mathcal{A}_{mu}$ is $sigma$-algebra and $mu$ can be uniquely extended from algebra $mathcal{A}$ to $sigma$-algebra $mathcal{A}_{mu}$. And this extension is given by $mu^*$.
I would be very grateful if somebody can detailed answer. Because it is important two understand such things.
EDIT: I will divide my questions into 3 parts.
1) Corollary 1.5.8. (page 21).
I was not able to understand this part. He just uses the definition of outer measure $mu^*$ but not the definition of $mu$-measurability.
My approach: Since $A$ is $mu$-measurable then $forall varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$ such that $mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using monotonicity of $mu^*$ we have $mu^*(A-A_{varepsilon})leq mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using subadditivity of $mu^*$ we have $mu^*(A)-mu^*(A_{varepsilon})leq mu^*(A-A_{varepsilon})<varepsilon.$ But since $A_{varepsilon}in mathcal{A}$ then $mu^*(A_{varepsilon})=mu(A_{varepsilon})$ so $mu^*(A)-mu(A_{varepsilon})<varepsilon$. We have that $mu^*(A)$ is finite so we can apply the definition of outer measure. Is it correct? But I am sure that it is OK.
2)
He takes $Bin mathcal{A}_{mu}$ but why $mu(B)$ makes sense? $mu(B)$ is defined only on $malcal{A}$ but $B$ may not be in $mathcal{A}$. This also confusing me a lot.
3)
The same question $B$ and $C$ are $mu$-measurable sets but why $mu(B)$ and $mu(C)$ makes sense?
I guess that all these questions are related to each. So would be very grateful for detailed answer and help!
EDIT 2: Also one moment which I have forgot to ask in the previous questions:
In the above proof note two moments which I have underlined with red line. Why Bogachev writes $mu^*(A)=mu^*(A'')$ but in he writes just $mu(B)=mu^*(X-A)$. The first is OK and I am agree with that but in the second he omits $mu^*$ and just write $mu(B)$. Could you explain it, please?
measure-theory
I am reading Bogachev's book "Measure Theory" which is in my opinion is very good book on measure theory. Let me ask you the following question:
Definition: Let $mu$ - non-negative set function with domain $mathcal{A}subset 2^X$. The set $A$ is called $mu$-measurable if
for any $varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$
such that $mu^*(Atriangle A_{varepsilon})<varepsilon,$ where
$mu^*$-outer measure.
The set of all $mu$-measurable sets is denoted by
$mathcal{A}_{mu}$.
One thing began to confuse me during reading. When author takes some set $Ain mathcal{A}_{mu}$ and then he considers $mu(A)$. This confuses me a lot.
We know that $mathcal{A}subset mathcal{A}_{mu}$ and when $Ain mathcal{A}_{mu}$ it means that $A$ is $mu$-measurable set. But $mu$ is defined only on $mathcal{A}$. Why he does not consider $mu^*(A)$?
More precisely, if $A$ is $mu$-measurable ($Ain mathcal{A}_{mu}$) then why $mu(A)$ makes sense? and does $mu(A)=mu^*(A)$?
P.S. I know the fact that $mathcal{A}_{mu}$ is $sigma$-algebra and $mu$ can be uniquely extended from algebra $mathcal{A}$ to $sigma$-algebra $mathcal{A}_{mu}$. And this extension is given by $mu^*$.
I would be very grateful if somebody can detailed answer. Because it is important two understand such things.
EDIT: I will divide my questions into 3 parts.
1) Corollary 1.5.8. (page 21).
I was not able to understand this part. He just uses the definition of outer measure $mu^*$ but not the definition of $mu$-measurability.
My approach: Since $A$ is $mu$-measurable then $forall varepsilon>0$ there exists $A_{varepsilon}in mathcal{A}$ such that $mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using monotonicity of $mu^*$ we have $mu^*(A-A_{varepsilon})leq mu^*(Atriangle A_{varepsilon})<varepsilon$. Then using subadditivity of $mu^*$ we have $mu^*(A)-mu^*(A_{varepsilon})leq mu^*(A-A_{varepsilon})<varepsilon.$ But since $A_{varepsilon}in mathcal{A}$ then $mu^*(A_{varepsilon})=mu(A_{varepsilon})$ so $mu^*(A)-mu(A_{varepsilon})<varepsilon$. We have that $mu^*(A)$ is finite so we can apply the definition of outer measure. Is it correct? But I am sure that it is OK.
2)
He takes $Bin mathcal{A}_{mu}$ but why $mu(B)$ makes sense? $mu(B)$ is defined only on $malcal{A}$ but $B$ may not be in $mathcal{A}$. This also confusing me a lot.
3)
The same question $B$ and $C$ are $mu$-measurable sets but why $mu(B)$ and $mu(C)$ makes sense?
I guess that all these questions are related to each. So would be very grateful for detailed answer and help!
EDIT 2: Also one moment which I have forgot to ask in the previous questions:
In the above proof note two moments which I have underlined with red line. Why Bogachev writes $mu^*(A)=mu^*(A'')$ but in he writes just $mu(B)=mu^*(X-A)$. The first is OK and I am agree with that but in the second he omits $mu^*$ and just write $mu(B)$. Could you explain it, please?
measure-theory
measure-theory
edited Nov 14 at 17:47
asked Nov 14 at 4:38
ZFR
4,88831337
4,88831337
1
Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53
add a comment |
1
Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53
1
1
Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
In 1) he does indeed not use the definition of $mu$-measurability, he refers to the definition of the outer measure.
2) He just started to denote the unique nonnegative countable additive extension $mu^*$ of $mu$ by $mu$ itself. Indeed, he does so already before definition 1.5.10 when he calls $(X,mathcal{A}_mu,mu)$ the Lebesgue completion of $(X,mathcal{A},mu)$. Using the same name for a function and an extension of it is a widespread harmless abuse of notation in much of mathematics.
3) Same as 2)
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
|
show 8 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
In 1) he does indeed not use the definition of $mu$-measurability, he refers to the definition of the outer measure.
2) He just started to denote the unique nonnegative countable additive extension $mu^*$ of $mu$ by $mu$ itself. Indeed, he does so already before definition 1.5.10 when he calls $(X,mathcal{A}_mu,mu)$ the Lebesgue completion of $(X,mathcal{A},mu)$. Using the same name for a function and an extension of it is a widespread harmless abuse of notation in much of mathematics.
3) Same as 2)
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
|
show 8 more comments
up vote
1
down vote
accepted
In 1) he does indeed not use the definition of $mu$-measurability, he refers to the definition of the outer measure.
2) He just started to denote the unique nonnegative countable additive extension $mu^*$ of $mu$ by $mu$ itself. Indeed, he does so already before definition 1.5.10 when he calls $(X,mathcal{A}_mu,mu)$ the Lebesgue completion of $(X,mathcal{A},mu)$. Using the same name for a function and an extension of it is a widespread harmless abuse of notation in much of mathematics.
3) Same as 2)
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
|
show 8 more comments
up vote
1
down vote
accepted
up vote
1
down vote
accepted
In 1) he does indeed not use the definition of $mu$-measurability, he refers to the definition of the outer measure.
2) He just started to denote the unique nonnegative countable additive extension $mu^*$ of $mu$ by $mu$ itself. Indeed, he does so already before definition 1.5.10 when he calls $(X,mathcal{A}_mu,mu)$ the Lebesgue completion of $(X,mathcal{A},mu)$. Using the same name for a function and an extension of it is a widespread harmless abuse of notation in much of mathematics.
3) Same as 2)
In 1) he does indeed not use the definition of $mu$-measurability, he refers to the definition of the outer measure.
2) He just started to denote the unique nonnegative countable additive extension $mu^*$ of $mu$ by $mu$ itself. Indeed, he does so already before definition 1.5.10 when he calls $(X,mathcal{A}_mu,mu)$ the Lebesgue completion of $(X,mathcal{A},mu)$. Using the same name for a function and an extension of it is a widespread harmless abuse of notation in much of mathematics.
3) Same as 2)
answered Nov 14 at 16:14
community wiki
Michael Greinecker
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
|
show 8 more comments
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
Could you take a look at my approach in question 1?
– ZFR
Nov 14 at 16:15
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
I am beginner in measure theory and that's why I am asking quite stupid questions. My approach explained to me why we can use the definition of outer measure. If the set $A$ is $mu$-measurable then it is outer measure, i.e. $mu^*(A)<infty$ and we can use the definition of outer measure (that what Bogachev did.)
– ZFR
Nov 14 at 16:18
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
Well, Bogachev starts with set functions whose values are real numbers. In particular, a nonnegative measure on algebra must be bounded above by the value of the total set $X$. He only starts to allow for infinite measures in 1.6.
– Michael Greinecker♦
Nov 14 at 16:22
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
And evry set has an outer measure.
– Michael Greinecker♦
Nov 14 at 16:24
1
1
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
No, he just mentions that the definition makes sense for set functions that can have the value $infty$, but that does not mean the results include such set functions. Indeed, not all results of this section hold true. The extension from a measure on an algebra to the generated $sigma$-algebra need for example not be unique. However, outer measure on the generated $sigma$-algebra is one such extension and if you want to allow for infinite measures, you need a proof that works for sets with infinite measure.
– Michael Greinecker♦
Nov 14 at 16:33
|
show 8 more comments
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Where exactly does he do what confuses you? Could you provide a more precise reference; this is a big book.
– Michael Greinecker♦
Nov 14 at 4:52
@MichaelGreinecker, Please take a look at the EDIT.
– ZFR
Nov 14 at 15:53