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Correspondence between dual of the space of continuous maps and signed measures
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On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found:
Let $(X, d)$ be a compact metric.
Recall that the dual space $C(X)^*$ of continuous real functionals on $C(X)$ of continuous functions $Xto mathbf R$ can be naturally identified with the space of finite signed measures on $X$ equipped with the weak* topology.
I want to confirm if the correspondence is given by the following map:
Let $mathcal F$ be the space of all the finite signed measures on $X$.
Define $phi:mathcal Fto C(X)^*$ as
$$phi(mu) = (fmapsto int f dmu)$$
measure-theory signed-measures
asked Jul 27 at 7:36
caffeinemachine
6,07221145
add a comment |Â
up vote
2
down vote
favorite
On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found:
Let $(X, d)$ be a compact metric.
Recall that the dual space $C(X)^*$ of continuous real functionals on $C(X)$ of continuous functions $Xto mathbf R$ can be naturally identified with the space of finite signed measures on $X$ equipped with the weak* topology.
I want to confirm if the correspondence is given by the following map:
Let $mathcal F$ be the space of all the finite signed measures on $X$.
Define $phi:mathcal Fto C(X)^*$ as
$$phi(mu) = (fmapsto int f dmu)$$
measure-theory signed-measures
asked Jul 27 at 7:36
caffeinemachine
6,07221145
1
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found:
Let $(X, d)$ be a compact metric.
Recall that the dual space $C(X)^*$ of continuous real functionals on $C(X)$ of continuous functions $Xto mathbf R$ can be naturally identified with the space of finite signed measures on $X$ equipped with the weak* topology.
I want to confirm if the correspondence is given by the following map:
Let $mathcal F$ be the space of all the finite signed measures on $X$.
Define $phi:mathcal Fto C(X)^*$ as
$$phi(mu) = (fmapsto int f dmu)$$
measure-theory signed-measures
asked Jul 27 at 7:36
caffeinemachine
6,07221145
On the first page of Chapter 4 of Einsiedler an Ward's Ergodic Theory: With a View Towards Number Theory, the following statement is found:
Let $(X, d)$ be a compact metric.
Recall that the dual space $C(X)^*$ of continuous real functionals on $C(X)$ of continuous functions $Xto mathbf R$ can be naturally identified with the space of finite signed measures on $X$ equipped with the weak* topology.
I want to confirm if the correspondence is given by the following map:
Let $mathcal F$ be the space of all the finite signed measures on $X$.
Define $phi:mathcal Fto C(X)^*$ as
$$phi(mu) = (fmapsto int f dmu)$$
measure-theory signed-measures
asked Jul 27 at 7:36
caffeinemachine
6,07221145
asked Jul 27 at 7:36
caffeinemachine
6,07221145
asked Jul 27 at 7:36
caffeinemachine
6,07221145
asked Jul 27 at 7:36
caffeinemachine
6,07221145
6,07221145
1
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19
add a comment |Â
1
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19
1
1
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19
add a comment |Â
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1
That is correct. This is, in fact, an isometric isomorphism if you endow measures with total variation norm.
– Kavi Rama Murthy
Jul 27 at 7:58
@KaviRamaMurthy Can you confirm one more thing? Is the weak* topology on $mathcal F$ (see OP) defined as follows: We say $mu_nto mu$ in the weak* topology if $int_X f dmu_nto int_X f dmu$ for all $fin C(X)$?
– caffeinemachine
Jul 27 at 17:25
Yes. Probabilists wrongly call this weak convergence, but it is weak* convergence.
– Kavi Rama Murthy
Jul 27 at 23:19