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edited Nov 30, 2009 at 11:08

Konstantin Slutsky

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In some sense you always have a natural topology that comes with a measure space. Here is what I mean by that.

Once you have a measure space you can associate an algebra of measurable sets with this space (you may want to factor this algebra by an ideal of sets o measure zero). This measure algebra, as any other Boolean algebra, has Stone Space associated to it. It is such a compact zero-dimensional topological space, that algebra of its clopen (closed and open) sets is isomorphic to the given Boolean algebra. You can then transfer measure to that Stone space to make it into a measure space (in general the measure will not be $\sigma$-additive, but you can recover it from the $\sigma$-additivity on the algebra).

This measure space is not isomorphic to the space we started from, but they have isomorphic measure algebras. Since in measure theory we care only about measure algebra we don't lose any information, but you now also have a topology on the space such that all clopen sets are measurable. In some natural cases all Baire sets will be measurable (for example, for Lebesgue spaces). So, it is probably more natural to consider sheaves on the Stone space of the measure algebra.

UPDATE: For completeness I decided to add some references. First of all there is a great (but in some places not easy to read) treatise by Fremlin Measure Theory. You can download all the volumes for free. Volume 3 contains a huge amount of information about measures on Boolean algebras. Another source is Vladimirov's book Boolean Algebras in Analysis (especially chapter 7, where he talks about Caratheodory construction on Boolean algebras).

In some sense you always have a natural topology that comes with a measure space. Here is what I mean by that.

Once you have a measure space you can associate an algebra of measurable sets with this space (you may want to factor this algebra by an ideal of sets o measure zero). This measure algebra, as any other Boolean algebra, has Stone Space associated to it. It is such a compact zero-dimensional topological space, that algebra of its clopen (closed and open) sets is isomorphic to the given Boolean algebra. You can then transfer measure to that Stone space to make it into a measure space (in general the measure will not be $\sigma$-additive, but you can recover it from the $\sigma$-additivity on the algebra).

This measure space is not isomorphic to the space we started from, but they have isomorphic measure algebras. Since in measure theory we care only about measure algebra we don't lose any information, but you now also have a topology on the space such that all clopen sets are measurable. In some natural cases all Baire sets will be measurable (for example, for Lebesgue spaces). So, it is probably more natural to consider sheaves on the Stone space of the measure algebra.

In some sense you always have a natural topology that comes with a measure space. Here is what I mean by that.

Once you have a measure space you can associate an algebra of measurable sets with this space (you may want to factor this algebra by an ideal of sets o measure zero). This measure algebra, as any other Boolean algebra, has Stone Space associated to it. It is such a compact zero-dimensional topological space, that algebra of its clopen (closed and open) sets is isomorphic to the given Boolean algebra. You can then transfer measure to that Stone space to make it into a measure space (in general the measure will not be $\sigma$-additive, but you can recover it from the $\sigma$-additivity on the algebra).

This measure space is not isomorphic to the space we started from, but they have isomorphic measure algebras. Since in measure theory we care only about measure algebra we don't lose any information, but you now also have a topology on the space such that all clopen sets are measurable. In some natural cases all Baire sets will be measurable (for example, for Lebesgue spaces). So, it is probably more natural to consider sheaves on the Stone space of the measure algebra.

UPDATE: For completeness I decided to add some references. First of all there is a great (but in some places not easy to read) treatise by Fremlin Measure Theory. You can download all the volumes for free. Volume 3 contains a huge amount of information about measures on Boolean algebras. Another source is Vladimirov's book Boolean Algebras in Analysis (especially chapter 7, where he talks about Caratheodory construction on Boolean algebras).

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created Nov 30, 2009 at 9:15

Konstantin Slutsky

1.3k
7
14

In some sense you always have a natural topology that comes with a measure space. Here is what I mean by that.

Once you have a measure space you can associate an algebra of measurable sets with this space (you may want to factor this algebra by an ideal of sets o measure zero). This measure algebra, as any other Boolean algebra, has Stone Space associated to it. It is such a compact zero-dimensional topological space, that algebra of its clopen (closed and open) sets is isomorphic to the given Boolean algebra. You can then transfer measure to that Stone space to make it into a measure space (in general the measure will not be $\sigma$-additive, but you can recover it from the $\sigma$-additivity on the algebra).

This measure space is not isomorphic to the space we started from, but they have isomorphic measure algebras. Since in measure theory we care only about measure algebra we don't lose any information, but you now also have a topology on the space such that all clopen sets are measurable. In some natural cases all Baire sets will be measurable (for example, for Lebesgue spaces). So, it is probably more natural to consider sheaves on the Stone space of the measure algebra.