- #1
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Suppose that the set of functions [itex]\{P^a:\mathcal S\rightarrow \mathcal L|\,a\in \mathcal L\}[/itex] has the property that for all [itex]s_1,s_2\in\mathcal S[/itex], [tex]s_1=s_2\ \Leftrightarrow\ \forall a\in \mathcal L~~ P^a(s_1)=P^a(s_2).[/tex] Suppose also that the following is true for each positive integer n:
Given [itex]s_1,\dots,s_n\in\mathcal S[/itex] and [itex]c_1,\dots,c_n\in[0,1][/itex] such that [itex]c_1+\cdots+c_n=1[/itex], there exists [itex]s_0\in\mathcal S[/itex] such that for all [itex]a\in\mathcal L[/itex], [tex]P^a(s_0)=\sum_{k=1}^n c_k P^a(s_k).[/tex] My problem is that I would like to think of [itex]\mathcal S[/itex] as some sort of structure rather than just a set (because I would like to know what an "automorphism" of [itex]\mathcal S[/itex] would be). So I'm wondering if the information given above (the existence and properties of the [itex]P^a[/itex] functions) is enough to implicitly define some sort of structure on [itex]\mathcal S[/itex]?
I would like to use the notation [itex]\sum_{k=1}^n c_k s_k[/itex] for [itex]s_0[/itex], and really think of this as a convex combination of the [itex]s_k[/itex]. Is there perhaps an abstract definition of "convex set" that doesn't even refer to a vector space? Maybe it's called something different, like "convex structure" or "convex space"? If there is such a definition, I think I would just like to show that the functions I've mentioned ensure that [itex]\mathcal S[/itex] is the underlying set of such a structure.
Maybe I should be trying to map [itex]\mathcal S[/itex] bijectively onto a convex subset of some vector space, and take the automorphisms to be vector space automorphisms restricted to that convex subset? Hm, that actually sounds good, but how do I do that, and how do I justify thinking of restrictions of vector space automorphisms as automorphisms of [itex]\mathcal S[/itex]?
It appears that the books I'm reading (which are doing almost the same thing that I'm trying to do, but with a slightly different goal) don't really try to address this issue at this stage, and wait until they've made several additional assumptions which finally allows them to identify the members of [itex]\mathcal S[/itex] with probability measures on [itex]\mathcal L[/itex] (which by then has been equipped with a partial order and found to be a bounded orthocomplemented lattice). Maybe I'll have to do something like that too. I'm just wondering if something can be said at this early stage.
Given [itex]s_1,\dots,s_n\in\mathcal S[/itex] and [itex]c_1,\dots,c_n\in[0,1][/itex] such that [itex]c_1+\cdots+c_n=1[/itex], there exists [itex]s_0\in\mathcal S[/itex] such that for all [itex]a\in\mathcal L[/itex], [tex]P^a(s_0)=\sum_{k=1}^n c_k P^a(s_k).[/tex] My problem is that I would like to think of [itex]\mathcal S[/itex] as some sort of structure rather than just a set (because I would like to know what an "automorphism" of [itex]\mathcal S[/itex] would be). So I'm wondering if the information given above (the existence and properties of the [itex]P^a[/itex] functions) is enough to implicitly define some sort of structure on [itex]\mathcal S[/itex]?
I would like to use the notation [itex]\sum_{k=1}^n c_k s_k[/itex] for [itex]s_0[/itex], and really think of this as a convex combination of the [itex]s_k[/itex]. Is there perhaps an abstract definition of "convex set" that doesn't even refer to a vector space? Maybe it's called something different, like "convex structure" or "convex space"? If there is such a definition, I think I would just like to show that the functions I've mentioned ensure that [itex]\mathcal S[/itex] is the underlying set of such a structure.
Maybe I should be trying to map [itex]\mathcal S[/itex] bijectively onto a convex subset of some vector space, and take the automorphisms to be vector space automorphisms restricted to that convex subset? Hm, that actually sounds good, but how do I do that, and how do I justify thinking of restrictions of vector space automorphisms as automorphisms of [itex]\mathcal S[/itex]?
It appears that the books I'm reading (which are doing almost the same thing that I'm trying to do, but with a slightly different goal) don't really try to address this issue at this stage, and wait until they've made several additional assumptions which finally allows them to identify the members of [itex]\mathcal S[/itex] with probability measures on [itex]\mathcal L[/itex] (which by then has been equipped with a partial order and found to be a bounded orthocomplemented lattice). Maybe I'll have to do something like that too. I'm just wondering if something can be said at this early stage.