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July 12th, 2011, 06:29 AM   #1
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Lebesgue Spaces in economics

So I am reading a quantifiable finance text book for fun (my other hobbies include basketball and running) and I have come across L spaces. Now I realize this book is over my head but its enjoyable so I will continue reading. It seems to me Lebesgue profitability spaces set of points corresponding to moments (i.e. L1-->E(X) L2-->Var(x) etc.) (Yes I looked up the more mathematical definition but this way I can place it in economics) But I cannot conceptualize L0 or L(infinity) probability space and the set of points they would encompass. Thanks,

Cameron

PS The textbook uses something like L0(omega,F,P) so I am really wondering in that context. I get in the rigourous mathamatical sense the both L0 and L(infinity) are limits but I am concerned and that L0 is not really rigorously defined but again I am only concerned about how they are used in a financial way although drawing parallels between the math and the finance would be nice.
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July 12th, 2011, 12:42 PM   #2
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Re: Lebesgue Spaces in economics

Lebesgue spaces are usually defined in the range (1,?), where L? means essentially bounded. I've never come across L spaces for p < 1.

Note that probability spaces are a subset, since in general, the total integral doesn't have to be 1, nor does the integrand have to be non-negative.
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July 13th, 2011, 07:57 AM   #3
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Re: Lebesgue Spaces in economics

L0 is not a legitimate L space because it doesn't satisfy the triangle inequality but I think they use the same formula integral(abs(x)^p)ds to define the space I just don't understand what the role of the L0 subspace is for economics what does that set (as well as L(infinity)) play for economics.
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July 13th, 2011, 03:17 PM   #4
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Re: Lebesgue Spaces in economics

I have no idea what the application of any Lp space is to economics.
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July 14th, 2011, 10:27 AM   #5
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Re: Lebesgue Spaces in economics

Quote:
Originally Posted by mathman
I have no idea what the application of any Lp space is to economics.
It looks to me as the original poster just defined a standard but biased way of estimating the expecation and variance of a quantity:
Quote:
(i.e. L1-->E(X) L2-->Var(x) etc.)
Of course one should ask why L2 should correspond to the variance rather than the mean square. Saying that Lp space relates to economics is like saying addition relates to economics.
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July 14th, 2011, 01:21 PM   #6
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Re: Lebesgue Spaces in economics

Quote:
Originally Posted by cameronfen
L0 is not a legitimate L space because it doesn't satisfy the triangle inequality but I think they use the same formula integral(abs(x)^p)ds to define the space I just don't understand what the role of the L0 subspace is for economics what does that set (as well as L(infinity)) play for economics.
Lp for p < 1 does not satisfy triangle inequality.
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