((si,t ),(s i+1, )]. (We shall use this notation in the other cases
as well.) We have
ZI m (R ) I) A (f) l VarR(f) AR(If[)
ij i,J ij
(cf. Remark 4 following Defn. 2.3.1), and the right hand side is just
mlfi(R). The family (R ,) forms a disjoint cover of R with
Ri R, so taking supremum we obtain Iml(R) g mL (R). For the
ij IN .
other inequality, let E>0. There exists a grill oxi such that
I A R (f)| > VarR(f) E m (R) .
1,j i,j
But the left hand side is equal to E Im (Rij ) and the Ri
i,jj
forms a decomposition of R, so we have
lmf(R)l a Z Jmf(R1'j) = E Z | (f)l > if ,(R) E,
iJ lj l mi
i.e., imf(R)I > m fl(R) E. Letting E + 0 (neither side now depends
on the corresponding grill), we obtain Imf(R)J a m fl(R); hence
Imf(R) = m f (R).
2) Suppose now (s,t) lies in the second quadrant, i.e., s