By M.M. Rao
Presents formerly unpublished fabric at the fundumental pronciples and homes of Orlicz series and serve as areas. Examines the pattern direction habit of stochastic tactics. offers functional functions in information and probability.
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Extra info for Applications Of Orlicz Spaces (Pure and Applied Mathematics)
See Hewitt and Stromberg , p. 225227. ) The (7-finiteness of (JL is no restriction here. Corollary 7. (Clarkson ) Suppose that 1 < p < oo, = -^, and (£), E, p,} is a a -finite measure space. )* < 2* ( H / I I 2 + W|)*, 2 < p < oo. (25) Proof. If 1 < p < 2, let 1 < a < p < 2,$(u) = |w| a ,$ 0 (w) = u2 and s = * I ° . Then 0 < s < 1 and ^ w = w^ or $ a u = u p . 4,1 ^ from (17). Similarly let 2 < p < b < oo, $(u) = |w|6, $ 0 (w) = ^2 and s = ^Ef}Then 0 < s < 1 and 4> s (n) = w p, lim -s = p.
The proof of this result is found in Rao and Ren (, p. 83), and the last statement uses the same argument as in the above book (p. 46). However, we include the proof that (iii) => (i) when p, is diffuse on a set r^ 0 ,0 < /i(^o) < CXD, to indicate the argument. Thus if (i) is not true, so that $ ^ A2(oo), then there exist un t oo )>n$(un),n>l, (7) such that $(MI)//(Q O ) > 1 and Gn € £(O 0 ), the trace cr-algebra, satisfying Q(2un}p,(Gn} = 1. Consider the function fn — unxG • Then by (7) 1 1 P3>(fn) ~ $(Un)P'(Gn} < — $(2,Un)p,(Gn) n = n > 0, U —>• OO.
In particular, if 17 is compact then the bounded functions are dense so that L°° is dense in M*. These statements follow from the above proposition and standard results of Real Analysis. For simplicity, we use the notations L® and L^ for the spaces (L*, || • ||$) and (I/5, || • ||($)) respectively. Similarly for M* and M^. In case n is a counting measure, the corresponding sequence spaces are denoted i*,t^ and m*,m^ respectively. We first present alternative formulas for the gauge and Orlicz norms, using Theorems 6 and 8.