Order Topology And Preference
Order Topology And Preference
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All we need to show is that the diagram ^ w(X) u(X) commutes for some function f, and for that it suffices to show that u is constant on the inverse image w (to) of each co e w(X), since sending f: (J l-> u(w~Va))) then defines f as the desired function. [If N were a finite set equal to, say, M = {O, ... , m}, then the constancy of u on w~''((jj) for each w c w(X) would be clear as a con- sequence of having a|b for each n e N. ] To show this in general, i. E. , for arbitrary N, we use transfini...te induction.
Thus, consider some well-ordering of N, and denote initial segments of elements n c N by N(ri). For each y e N, define B = U B and C = A \ B . ^ N(y) ^ ^' ^ and the functions w and w on X by w^(x) = (x^*, {ug^ (-B^)^neN(y)^ ^"'^ w (x) = (X, u (X ), {u (xg )}^, N(y)^' where C^ = C^\B^.
Now we suppose that u is constant on -29- p e V (X ) for each n e N, in the sense that (a) if n n B^ P^ > P^ c ^n^B ^ ^°^ ^^^^ n e N, then f (x^, p) > f(x^, p') for each x^ e X, and (b) if the hypothesis of (a) holds with, furthermore, p, > p' for some A e N, then f(X|.
What to read after Order Topology And Preference?
You can find similar books in the "Read Also" column, or choose other free books by Murat R Sertel to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
All we need to show is that the diagram ^ w(X) u(X) commutes for some function f, and for that it suffices to show that u is constant on the inverse image w (to) of each co e w(X), since sending f: (J l-> u(w~Va))) then defines f as the desired function. [If N were a finite set equal to, say, M = {O, ... , m}, then the constancy of u on w~''((jj) for each w c w(X) would be clear as a con- sequence of having a|b for each n e N. ] To show this in general, i. E. , for arbitrary N, we use transfini...te induction.
Thus, consider some well-ordering of N, and denote initial segments of elements n c N by N(ri). For each y e N, define B = U B and C = A \ B . ^ N(y) ^ ^' ^ and the functions w and w on X by w^(x) = (x^*, {ug^ (-B^)^neN(y)^ ^"'^ w (x) = (X, u (X ), {u (xg )}^, N(y)^' where C^ = C^\B^.
Now we suppose that u is constant on -29- p e V (X ) for each n e N, in the sense that (a) if n n B^ P^ > P^ c ^n^B ^ ^°^ ^^^^ n e N, then f (x^, p) > f(x^, p') for each x^ e X, and (b) if the hypothesis of (a) holds with, furthermore, p, > p' for some A e N, then f(X|.
What to read after Order Topology And Preference?
You can find similar books in the "Read Also" column, or choose other free books by Murat R Sertel to read onlineMoreLess
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