By Jacques Hadamard
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Additional resources for An essay on the psychology of invention in the mathematical field
E. is a partial order on L4,. e. e. If f, g E L4,, then the pointwise maximum f V g and pointwise minimum f A g also belong to L4,. In particular, the pointwise absolute value If I belongs to L4, and II I f I 14 = I I f 114. Observe that these closure properties are true as well for the heart H4,. It is also a Banach lattice. 11(c)) may fail for H4,, however. ] The space L4, is a rearrangement invariant function space. This means that if f E L4, and g is a rearrangement of f, then g E L4, and II9II4, = 1 1 1 1 14, .
Then J is directed by refinement. ) Then for t E J, let Xt = µ(A) 1A, P(A) where the sum is over all A E t with P(A) # 0. Then (Xt) is an Llbounded supermartingale, so it converges stochastically, say to Y. Then pa(B) = E [Y 1B], B E 9, and u,(B) = p(B) -pa(B) define the required decomposition. For general p, use the finiteness of the variation of µ to write p as the difference of two nonnegative charges. Ft). 16). 16) to the charge µ(A) = limt E [Xt 1A], for A E 9 = Ut F. 1); then clearly Zt -+ 0 stochastically.
16b) -1)(2u) < kd)(u) for 0 < u < uo. 16a) requires that 4i(u) < oo for all u. 16b) requires that 1(u) > 0 for all u > 0. 1. 17) Theorem. Let (1k, F, µ) be a a-finite measure space, and let be an Orlicz function. (1) If 4) satisfies (02) at 0 and oo, then H4,(1l, F, µ) = Lq, (1, F, µ). , ([0, oo)), then 4) satisfies (02) at 0 and oo. (2) Suppose p is finite. If 4) satisfies (02) at oo, then H4, (SZ,F, p) _ Lj (SZ, F, µ). If 114 ([0,1]) = L4 ([0,1]), then 4) satisfies (02) at 00. (3) Suppose u has no arbitrarily small sets.
An essay on the psychology of invention in the mathematical field by Jacques Hadamard