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 July 12th, 2011, 06:29 AM #1 Newbie   Joined: Jul 2011 Posts: 2 Thanks: 0 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.
 July 12th, 2011, 12:42 PM #2 Global Moderator   Joined: May 2007 Posts: 6,764 Thanks: 697 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.
 July 13th, 2011, 07:57 AM #3 Newbie   Joined: Jul 2011 Posts: 2 Thanks: 0 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.
 July 13th, 2011, 03:17 PM #4 Global Moderator   Joined: May 2007 Posts: 6,764 Thanks: 697 Re: Lebesgue Spaces in economics I have no idea what the application of any Lp space is to economics.
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.

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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