63.8. LEVY’S THEOREM 2171

complements. Let K denote those sets which are finite intersections of sets of the form(X (u)−X (r))−1 (B) where B is a Borel set and r≤ u≤ s. Say a set, A of K is of the form

∩mi=1 (X (ui)−X (ri))

−1 (Bi)

Then since disjoint increments are independent, linear combinations of the random vari-ables, X (ui)−X (ri) are normally distributed. Consequently,

(X (u1)−X (r1) , · · · ,X (um)−X (rm) ,X (t)−X (s))

is multivariate normal. The covariance matrix is of the form(A 00 t− s

)and so the random vector, (X (u1)−X (r1) , · · · ,X (um)−X (rm)) and the random variableX (t)−X (s) are independent. Consequently, XA is independent of X (t)−X (s) for anyA ∈ K . Then by the lemma on π systems, Lemma 12.12.3 on Page 329, Fs ⊇ G ⊇σ (K ) = Fs. This proves the claim.

Thus ∫A(X (t)−X (s))dP =

∫Ω

(X (t)−X (s))XAdP

= P(A)∫

Ω

(X (t)−X (s))dP = 0

which shows that since A ∈Fs was arbitrary,

E (X (t) |Fs) = X (s)

and {X (t)} is a martingale.Now consider whether

{X (t)2− t

}is a martingale. By assumption,

L (X (t)−X (s)) = L (X (t− s)) = N (0, t− s) .

Then for A ∈Fs, the independence of XA and X (t)−X (s) shows∫A

E((X (t)−X (s))2 |Fs

)dP =

∫A(X (t)−X (s))2 dP

= P(A)(t− s) =∫

A(t− s)dP

and since A ∈Fs is arbitrary,

E((X (t)−X (s))2 |Fs

)= t− s

and so the result follows from Lemma 63.8.2. This proves the theorem.The next lemma is the main result from which Levy’s theorem will be established.