- THE FUNCTIONALSμ AND v
NowHenceII^9 klvk - (<I>J;l)*^900 lloe(wk(supp(f)),(w;;^1 )*goo)= llk [^9 klvk] -^900 lloe(supp(f),g 00 )
---+ o.Ilk (41fT)-n/2e-fo<f?kl dμ(w;;l )*goo - lk (47rT)-n/2e-fo<f?k1 dμgk I
S (47rT)-n/2 r e-fo<f?kl (1 - dμgk ) dμ -1 *
} N; k dμ(.._-1)* "'k goo (<Pk ) g^00S 119klvk - (<I>J;l)* 9ooll~~<r?k(supp(f)),(w;;l)*goo)X r (41fT)-n/2e-fo<f?kl dμ(w-1)*
}Nk k goo
---+ 0,which impliesWe conclude thatW (Noo, 900, f, T) =kl~ W (Uk, <I>k [ 9klvk] , f, T)= lim W (Nki 9k, f o <I>J;1, T)
k-->oo
2: limsup μ (9k, T).
k-->oo241To see why the last inequality is true, note that the functions fk ~ f o J;^1 +
log ck satisfy ck ---+ 1 and the constraints
andr (41fT)-nf^2 e-fkdμgk = 1
}Nkw (Nk, 9k, Jo J;^1 , T) = ckw (Nk, 9k, fk, T) - ck log ck
2: Ckμ (9k, T) - ck log ck.
Since e-f/^2 is an arbitrary nonnegative function with compact support and
since it satisfies the constraint with respect to 900 , we obtain
μ(9 00 ,T) 2: limsupμ(9k,T).
k-->oo
D