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corrected some irritating typos.
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jbc
jbc
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This more a comment than an answer but will be too long for that. There are tootwo basic approaches to the theory of finite measures---the topogical one and the one based on $\sigma$-algebras. At the core of the first approach lies the duality between $C^b(S)$ (the bounded, continuous functions on a completely regular space $S$) and the space $M^t(S)$ of Radon measures thereon, for the second that between $L^\infty(\Omega)$ (the bounded measurable functions on a measure space $(\Omega,\cal A)$ and the space $\cal M(\cal A)$ of finite $\sigma$-additive real valued-measures. All four of these spaces are Banach spaces in a natural way but these structures are not compatible with the above dualities. This suggests that one should try to provide them withthe function spaces with suitable locally convex topologies which are compatible. In the case of $C^b(S)$ this means a suitable intrinsic locally convex topology which is complete and has the required dual space $M^t(S))$. Due to the joint efforst of several mathematicians (in particular, R.C. Buck and the polish school, e.g., Orlicz and Wiweger) such a topology is known---it is the so-called strict topology (Strictlystrictly speaking, this is not always complete, but it is so for most of the standard classes of topological spaces).

There is such a topology in the second case but I have been unable to find it in the literature which isand this is the reason that I am submitting this answer. It has various descriptions (this inin itself is, in my opinion, a hint that it is a useful and natural topology). But for me the killer-diller fact is that it has the natural universal property for $\sigma$-additive measures with values in a Banach space (in contrast to the topological case, where we can embed $S$ into the space $M^t(S)$ with corresponding universal property), we can here embed the $\sigma$-algebra $\cal A$ into $L^\infty(\Omega)$ in the natural way.

We close with a few remarks.

$1$. We emphasise that it is important that the above topologies have intrinsic definitions (i..e., independent of the dual pairs). The fact that they then turn out to be topologies which can be defined via the duality---typically the Mackey topology---is then a theorem to be proved. This is, basically, the reason for the relevance of all this to the OP which essentially asks for a characterisations of the weakly compact subsets of of the the dual of $L^\infty$.

$2$. Another reason for depositing this was to clarify the confusion about which is the relevant topology on the family of measures---it is clear from the formulation that the OP refers to the weak star topologtopology of a subsset of the dual of $L^\infty$ with the topology referred to above.

$3$. If one tries to weaken the topology of a Banach space in a non-trivial way (as we are doing here) then there are contraints on the kind of space one can get. For example, one cannot get a barrelled space (closed graph theorem). This means that the resulting space topology cannot be a member of one of the traditional classes of well-behaved spacesLCS's (metrisable, nice inductive liitslimits of Banach spaces, e.g.). They also cannot beNeither can they be nuclear (except, of course, in trivial cases). This, alas, seems to be ahave been a hindrance to their acceptance into the main body of functional analysis. It is my belief that this is a great pity since they are precisely the tool required for extending and enriching the relationship between functional analysis (duality theory) and measure theory (the Riesz representation theorem) which is one of the crown jewels of analysis.

This more a comment than an answer but will be too long for that. There are too basic approaches to the theory of finite measures---the topogical one and the one based on $\sigma$-algebras. At the core of the first approach lies the duality between $C^b(S)$ (the bounded, continuous functions on a completely regular space $S$) and the space $M^t(S)$ of Radon measures thereon, for the second that between $L^\infty(\Omega)$ (the bounded measurable functions on a measure space $(\Omega,\cal A)$ and the space $\cal M(\cal A)$ of finite $\sigma$-additive real valued-measures. All four of these spaces are Banach spaces in a natural way but these are not compatible with the above dualities. This suggests that one should try to provide them with suitable locally convex topologies which are compatible. In the case of $C^b(S)$ this means a suitable intrinsic locally convex topology which is complete and has the required dual space $M^t(S))$. Due to the joint efforst of several mathematicians (in particular, R.C. Buck and the polish school, e.g., Orlicz and Wiweger) such a topology is known---it is the so-called strict topology (Strictly speaking, this is not always complete, but it is so for most of the standard classes of topological spaces).

There is such a topology in the second case but I have been unable to find it in the literature which is the reason that I am submitting this answer. It has various descriptions (this in itself is, in my opinion, a hint that it is a useful and natural topology). But for me the killer-diller fact is that it has the natural universal property for $\sigma$-additive measures with values in a Banach space (in contrast to the topological case, where we can embed $S$ into the space $M^t(S)$ with corresponding universal property), we can here embed the $\sigma$-algebra $\cal A$ into $L^\infty(\Omega)$ in the natural way.

We close with a few remarks.

$1$. We emphasise that it is important that the above topologies have intrinsic definitions (i..e. independent of the dual pairs). The fact that they then turn out to be topologies which can be defined via the duality---typically the Mackey topology---is then a theorem to be proved. This is, basically, the reason for the relevance of all this to the OP which essentially asks for a characterisations of the weakly compact subsets of of the dual of $L^\infty$.

$2$. Another reason for depositing this was to clarify the confusion about which is the relevant topology---it is clear from the formulation that the OP refers to the weak star topolog of a subsset of the dual of $L^\infty$ with the topology referred to above.

$3$. If one tries to weaken the topology of a Banach space in a non-trivial (as we are doing here) then there are contraints on the kind of space one can get. For example, one cannot get a barrelled space (closed graph theorem). This means that the resulting space cannot be a member of the traditional classes of well-behaved spaces (metrisable, nice inductive liits of Banach spaces, e.g.). They also cannot be nuclear (except, of course, in trivial cases). This, alas, seems to be a hindrance to their acceptance into the main body of functional analysis. It is my belief that this is a great pity since they are precisely the tool required for extending and enriching the relationship between functional analysis (duality theory) and measure theory (the Riesz representation theorem) which is one of the crown jewels of analysis.

This more a comment than an answer but will be too long for that. There are two basic approaches to the theory of finite measures---the topogical one and the one based on $\sigma$-algebras. At the core of the first approach lies the duality between $C^b(S)$ (the bounded, continuous functions on a completely regular space $S$) and the space $M^t(S)$ of Radon measures thereon, for the second that between $L^\infty(\Omega)$ (the bounded measurable functions on a measure space $(\Omega,\cal A)$ and the space $\cal M(\cal A)$ of finite $\sigma$-additive real valued-measures. All four of these spaces are Banach spaces in a natural way but these structures are not compatible with the above dualities. This suggests that one should try to provide the function spaces with suitable locally convex topologies which are compatible. In the case of $C^b(S)$ this means a suitable intrinsic locally convex topology which is complete and has the required dual space $M^t(S))$. Due to the joint efforst of several mathematicians (in particular, R.C. Buck and the polish school, e.g., Orlicz and Wiweger) such a topology is known---it is the so-called strict topology (strictly speaking, this is not always complete, but it is so for most of the standard classes of topological spaces).

There is such a topology in the second case but I have been unable to find it in the literature and this is the reason that I am submitting this answer. It has various descriptions (in itself, in my opinion, a hint that it is a useful and natural topology). But for me the killer-diller fact is that it has the natural universal property for $\sigma$-additive measures with values in a Banach space (in contrast to the topological case, where we can embed $S$ into the space $M^t(S)$ with corresponding universal property), we can here embed the $\sigma$-algebra $\cal A$ into $L^\infty(\Omega)$ in the natural way.

We close with a few remarks.

$1$. We emphasise that it is important that the above topologies have intrinsic definitions (i.e., independent of the dual pairs). The fact that they then turn out to be topologies which can be defined via the duality---typically the Mackey topology---is then a theorem to be proved. This is, basically, the reason for the relevance of all this to the OP which essentially asks for a characterisations of the weakly compact subsets of the dual of $L^\infty$.

$2$. Another reason for depositing this was to clarify the confusion about which is the relevant topology on the family of measures---it is clear from the formulation that the OP refers to the weak star topology of a subsset of the dual of $L^\infty$ with the topology referred to above.

$3$. If one tries to weaken the topology of a Banach space in a non-trivial way (as we are doing here) then there are contraints on the kind of space one can get. For example, one cannot get a barrelled space (closed graph theorem). This means that the resulting topology cannot be a member of one of the traditional classes of well-behaved LCS's (metrisable, inductive limits of Banach spaces, e.g.). Neither can they be nuclear (except, of course, in trivial cases). This, alas, seems to have been a hindrance to their acceptance into the main body of functional analysis. It is my belief that this is a great pity since they are precisely the tool required for extending and enriching the relationship between functional analysis (duality theory) and measure theory (the Riesz representation theorem) which is one of the crown jewels of analysis.

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jbc
jbc
  • 2.4k
  • 1
  • 26
  • 11

This more a comment than an answer but will be too long for that. There are too basic approaches to the theory of finite measures---the topogical one and the one based on $\sigma$-algebras. At the core of the first approach lies the duality between $C^b(S)$ (the bounded, continuous functions on a completely regular space $S$) and the space $M^t(S)$ of Radon measures thereon, for the second that between $L^\infty(\Omega)$ (the bounded measurable functions on a measure space $(\Omega,\cal A)$ and the space $\cal M(\cal A)$ of finite $\sigma$-additive real valued-measures. All four of these spaces are Banach spaces in a natural way but these are not compatible with the above dualities. This suggests that one should try to provide them with suitable locally convex topologies which are compatible. In the case of $C^b(S)$ this means a suitable intrinsic locally convex topology which is complete and has the required dual space $M^t(S))$. Due to the joint efforst of several mathematicians (in particular, R.C. Buck and the polish school, e.g., Orlicz and Wiweger) such a topology is known---it is the so-called strict topology (Strictly speaking, this is not always complete, but it is so for most of the standard classes of topological spaces).

There is such a topology in the second case but I have been unable to find it in the literature which is the reason that I am submitting this answer. It has various descriptions (this in itself is, in my opinion, a hint that it is a useful and natural topology). But for me the killer-diller fact is that it has the natural universal property for $\sigma$-additive measures with values in a Banach space (in contrast to the topological case, where we can embed $S$ into the space $M^t(S)$ with corresponding universal property), we can here embed the $\sigma$-algebra $\cal A$ into $L^\infty(\Omega)$ in the natural way.

We close with a few remarks.

$1$. We emphasise that it is important that the above topologies have intrinsic definitions (i..e. independent of the dual pairs). The fact that they then turn out to be topologies which can be defined via the duality---typically the Mackey topology---is then a theorem to be proved. This is, basically, the reason for the relevance of all this to the OP which essentially asks for a characterisations of the weakly compact subsets of of the dual of $L^\infty$.

$2$. Another reason for depositing this was to clarify the confusion about which is the relevant topology---it is clear from the formulation that the OP refers to the weak star topolog of a subsset of the dual of $L^\infty$ with the topology referred to above.

$3$. If one tries to weaken the topology of a Banach space in a non-trivial (as we are doing here) then there are contraints on the kind of space one can get. For example, one cannot get a barrelled space (closed graph theorem). This means that the resulting space cannot be a member of the traditional classes of well-behaved spaces (metrisable, nice inductive liits of Banach spaces, e.g.). They also cannot be nuclear (except, of course, in trivial cases). This, alas, seems to be a hindrance to their acceptance into the main body of functional analysis. It is my belief that this is a great pity since they are precisely the tool required for extending and enriching the relationship between functional analysis (duality theory) and measure theory (the Riesz representation theorem) which is one of the crown jewels of analysis.