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 Post subject: Measure preservingPosted: Sat, 28 Jul 2012 22:50:27 UTC
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If I'm trying to show some action is measure preserving on a Borel -algebra, is it enough to show it on the basic open sets? Or, more generally, is it enough to show it on any system of generators?

If not, then what if I add any combination of the assumptions of regularity or finite measure? I feel like that *should* be closer to reasonable as the measure is continuous then, and I should only have to show it plays well with finite intersections and arbitrary unions of basic open sets since I can then take the inf over open sets containing my set.

Any help appreciated.

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 Post subject: Re: Measure preservingPosted: Sun, 29 Jul 2012 02:22:31 UTC
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If I'm trying to show some action is measure preserving on a Borel -algebra, is it enough to show it on the basic open sets? Or, more generally, is it enough to show it on any system of generators?

If not, then what if I add any combination of the assumptions of regularity or finite measure? I feel like that *should* be closer to reasonable as the measure is continuous then, and I should only have to show it plays well with finite intersections and arbitrary unions of basic open sets since I can then take the inf over open sets containing my set.

Any help appreciated.

Depends on what you mean by generators. For example, generates the Borel -algebra on (i.e. ), but the zero measure and the Dirac delta agree on .

If the measure is -finite, then by the uniqueness of extension (Caratheodory/Hahn-Kolmogorov), it is enough to show it on any -system generating , e.g. all open intervals in .

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 Post subject: Re: Measure preservingPosted: Mon, 30 Jul 2012 00:20:29 UTC
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I see, that's because -finiteness allows us to break up the sets into disjoint pieces we can add up, and continuity of the measure?

Still, I'm not quite sure why a system is enough, shouldn't there be closure under (relative) compliments as well? After all, {open sets not containing 0} are a -system, are they not? It seems like you're appealing to the Caratheodory theorem, but that needs the compliment bit as well, or I'm misunderstanding where you're coming from on that.

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 Post subject: Re: Measure preservingPosted: Mon, 30 Jul 2012 03:48:02 UTC
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I see, that's because -finiteness allows us to break up the sets into disjoint pieces we can add up, and continuity of the measure?

Still, I'm not quite sure why a system is enough, shouldn't there be closure under (relative) compliments as well? After all, {open sets not containing 0} are a -system, are they not? It seems like you're appealing to the Caratheodory theorem, but that needs the compliment bit as well, or I'm misunderstanding where you're coming from on that.

Oops, I don't mean -finiteness on , but -finite using sets in the pi-system , i.e. there exists such that (that disqualifies the {open sets not containing 0} example because you can never cover 0).

-finiteness allows us to add up and subtract, so we can get relative complements:
etc (so you can enlarge by assuming are nested) and
(for ).
Alternatively, we can also appeal to the uniqueness of (if two finite measures agree on a pi-system and have the same total mass, then Dynkin => they agree on the sigma-algebra generated) and take limit

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 Post subject: Re: Measure preservingPosted: Mon, 30 Jul 2012 03:54:39 UTC
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outermeasure wrote:
I see, that's because -finiteness allows us to break up the sets into disjoint pieces we can add up, and continuity of the measure?

Still, I'm not quite sure why a system is enough, shouldn't there be closure under (relative) compliments as well? After all, {open sets not containing 0} are a -system, are they not? It seems like you're appealing to the Caratheodory theorem, but that needs the compliment bit as well, or I'm misunderstanding where you're coming from on that.

Oops, I don't mean -finiteness on , but -finite using sets in the pi-system , i.e. there exists such that (that disqualifies the {open sets not containing 0} example because you can never cover 0).

-finiteness allows us to add up and subtract, so we can get relative complements:
etc (so you can enlarge by assuming are nested) and
(for ).

I'm just a little confused though, what does -finiteness have to do with whether or not the -system covers the space? I get what you meant to say now, this is just a question on terminology.

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 Post subject: Re: Measure preservingPosted: Mon, 30 Jul 2012 04:14:30 UTC
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outermeasure wrote:
I see, that's because -finiteness allows us to break up the sets into disjoint pieces we can add up, and continuity of the measure?

Still, I'm not quite sure why a system is enough, shouldn't there be closure under (relative) compliments as well? After all, {open sets not containing 0} are a -system, are they not? It seems like you're appealing to the Caratheodory theorem, but that needs the compliment bit as well, or I'm misunderstanding where you're coming from on that.

Oops, I don't mean -finiteness on , but -finite using sets in the pi-system , i.e. there exists such that (that disqualifies the {open sets not containing 0} example because you can never cover 0).

-finiteness allows us to add up and subtract, so we can get relative complements:
etc (so you can enlarge by assuming are nested) and
(for ).

I'm just a little confused though, what does -finiteness have to do with whether or not the -system covers the space? I get what you meant to say now, this is just a question on terminology.

It has nothing to do there, except I was thinking about the general statement of uniqueness of measure extended from pi-system to its sigma-algebra:

Theorem: Let be measures on a measurable space . Suppose that is a -system on generating such that on and that there exists a sequence with and . Then .
Proof: Apply Dynkin's theorem to . Details omitted.

The non-uniqueness when you don't have is demonstrated by , , when any multiple of counting measure will give the same result (infinity on nonempty) on .

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 Post subject: Re: Measure preservingPosted: Mon, 30 Jul 2012 04:23:35 UTC
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outermeasure wrote:
outermeasure wrote:
I see, that's because -finiteness allows us to break up the sets into disjoint pieces we can add up, and continuity of the measure?

Still, I'm not quite sure why a system is enough, shouldn't there be closure under (relative) compliments as well? After all, {open sets not containing 0} are a -system, are they not? It seems like you're appealing to the Caratheodory theorem, but that needs the compliment bit as well, or I'm misunderstanding where you're coming from on that.

Oops, I don't mean -finiteness on , but -finite using sets in the pi-system , i.e. there exists such that (that disqualifies the {open sets not containing 0} example because you can never cover 0).

-finiteness allows us to add up and subtract, so we can get relative complements:
etc (so you can enlarge by assuming are nested) and
(for ).

I'm just a little confused though, what does -finiteness have to do with whether or not the -system covers the space? I get what you meant to say now, this is just a question on terminology.

Ah, OK, then my hunch was right. Thank you very much for confirming that.
It has nothing to do there, except I was thinking about the general statement of uniqueness of measure extended from pi-system to its sigma-algebra:

Theorem: Let be measures on a measurable space . Suppose that is a -system on generating such that on and that there exists a sequence with and . Then .
Proof: Apply Dynkin's theorem to . Details omitted.

The non-uniqueness when you don't have is demonstrated by , , when any multiple of counting measure will give the same result (infinity on nonempty) on .

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