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Additional info for 2-transitive abstract ovals of odd order
On t he ot her han d, E, rt ~o (i = 1,2 , .. 8) we obtain nUe, rt 00 E = ( ~)n from above (Defini- 00 ~o . k= l i=k Since JV C JYO we have E rt JV . Since E E A we obt ain a contradiction 0 with the inclusion A C JV . 4 Small systems and convergence If (X, Y') is a measurable space and (~)n is a small system on Y', then the induced o-ideal J'V = n~=l ~ enables us to speak about the almost everywhere convergence (more precisely J'V-almost everywhere), or about J'V-almost everywhere uniform convergence.
0 i=1 = 1,... ri} , th en b r. (L~=l Ck) ~ = 1) we get Ck ) k=2 k=l =11=1 2k n i 1 =1 b /\ t 2k + c,) V (t, t, c,+ ~ (b /\ 2Cl) V (b /\ ( 2C2 + t k=3 Ck) ) 2Ck) ) (t, t, ~ (b A2c,) V(b A((2 C, + 2c,) V 2c,+ 2Ck) )9) c. 2 RANGE = (b/\ 2cd V (b /\ 22C2 ) V 45 (b /\ ( 2 2C3 + t 22Ck ) ) k=4 n ~ ... ~ V(b/\ ( 2 k Ck ) ) ~ C. 3. 6 00 ~ V at ,cp(t) = C. 7 n a /\ 00 LV 00 ak,i,cp(i+ k- l) ~ k=li = l V ai,cp(i) . i= l By the a-co nt inuity of G 00 a /\ 00 LV k=l i =l 00 ak ,i, cp(i+k - l) ~ V ai, cp(i) .
The ordering ~ has the usual sense. 5 Lemma. , E A (n = 1,2, .. ), an /" a, bn /" b, a ~ band (Jo(an))n and (Jo(bn))n are bounded, then VJo(an) ~ VJo(bn). n n Proof. By the assumption (3) we have an 1\ bm /" an 1\ b = an (m -+ 00), hence by (i) and (v) Jo(a n) = VJo(an 1\ bm) ~ VJo(bm). 6 Definition. By A+ we denote the set of all bE X such that there is a sequence (an)n of elements of A with an /" band (Jo(an) )n bounded. Further, we define a mapping J+: A + -+ G by the formula J+(b) = VJo(an), n where an E A , an /" b.