1 2 1 1
convention F = F = F (which may or may not equal F = VF or
F2 = VF2)
t 1 2
We then set Ft = F F and verify that this family satisfies
st s
the usual conditions.
1 2
1) Each Fst contains the P-negligible sets, since Fs, F contain
them for all s,t.
2) If (s,t) S (s',t'), then Ft C F t..
1 1 2 2
In fact, F1 C F since sSs', F2 C F, since t5t', hence
s s t t
1 2 1 F 2
Fst F (- F Fl C F OF Fs
st s t S t st
3) We have F = ( F ,. (Note: In two parameters,
st st
(s,t)<(s',t')
our usual definition of right limit is for (s,t) S (s',t'),
(s,t) (s',t'), as we shall see later. The statement we
shall prove is somewhat stronger.)
Proof. One containment is evident: since FstC Fs t, for
each (s',t') > (s,t) by (2), Fs C f Fst
t (s,t)<(s',t')
For the other containment, let
A c n F = (F nFi).
(s,t)<(s',t') s'>s
t'>t
Then A E F for all s'>s, hence A E Sn F = F
SS '>s
Similarly, A c F2 for all t'>t, hence A E } F2 = F2
t s
t'>t
Putting the two containments together gives the equality.