0 0
E(1A(V z- V )) for all A c F, so 1B B | S V V P-a.s. for
each z < z'. The same computations for the rectangle yield
o
A R (B ) S R (V) P-a.s. If we take z,z' with rational
zz zz
o
coordinates, we can find a common negligible set and modify B
on it to get the inequalities everywhere for all z,z' rational.
Next, let z be fixed. We show that for any sequence r z, r
n n
o
rational, the sequence (Br (w)) is Cauchy for all w. In fact, the
n
sequence (Vr (w)) is Cauchy for all w since V is right continuous,
n
i.e., for any )>0, there exists n such that n,m > n implies
IVr (w) Vr (w)J < E. Then, for n,m > n we have IB (w) Br (w)
n m n m
0
SIVr (w) Vr (w)I < E. Thus, for any w, the sequence (B (w)) Is
n m n
0
Cauchy in L(E,Z') complete, so lim B (w) exists.
n n
Now, let (rn),(sn ) be two sequences of rationals decreasing to
z. We can construct a sequence (v ) decreasing to z, containing
0
subsequences of both (rn ) and (s ). Then lim B (w) exists; moreover,
nJ J
0 0
since llm Br (w) and lim Bs (w) exist, and subsequences of both are
n n n
0
contained In (B (w)), all three limits are equal. In particular,
0 0
lim Br (w) = lim Bs (w), so we get the same limit for any sequence of
n n n n
2
rational decreasing to z. Then, for every w e 0, z R ,
o
Bz(w) = lim B (w) exists. The stochastic function B thus
r+z
r rational
defined is right continuous.