Elect. Comm. in Probab.15(2010), 189–191 ELECTRONIC COMMUNICATIONS in PROBABILITY
CORRECTION TO “THE MEASURABILITY OF HITTING TIMES”
RICHARD F. BASS1
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 email: [email protected]
SubmittedFebruary 7, 2011, accepted in final formMarch 15, 2011 AMS 2000 Subject classification: Primary: 60G07; Secondary: 60G05, 60G40
Keywords: Stopping time, hitting time, progressively measurable, optional, predictable, debut theorem, section theorem
Abstract
We correct an error in[1].
There is an error in the proof of Theorem 2.1, which was pointed out to me by K. Szczypkowski. On page 101, line 18, althoughAn\Ln⊂ ∪n
i=1(Ai\Ki), the assertion concerning the projections is
not necessarily true.
The following should replace the proof of Theorem 2.1, from line 6 of page 101 through the line that is 7 lines before the end of page 101.
A
The correction
A setAis aP-null set ifP∗(A) =0. The following lemma is well known; see, e.g.,[2, p. 94]. Lemma A.1. (a) If A⊂Ω, there exists C∈ F such that A⊂C andP∗(A) =P(C).
(b) Suppose An↑A. ThenP∗(A) =limn→∞P∗(An).
Proof.(a) By the definition ofP∗(A), for eachnthere existsCn∈ F such thatA⊂CnandP(Cn)≤
P∗(A) + (1/n). SettingC=∩nCn, we haveA⊂C,C∈ F, andP(C)≤P(Cn)≤P∗(A) + (1/n)for
eachn, henceP(C)≤P∗(A).
(b) ChooseCn∈ F withAn⊂CnandP∗(An) =P(Cn). Let Dn=∩k≥nCk andD=∪nDn. We see
thatDn↑D,D∈ F, andA⊂D. Then
P∗(A)≥sup
n P
∗(
An) =sup n P
(Cn)≥sup n P
(Dn) =P(D)≥P∗(A).
Let Tt = [0,t]×Ω. Given a compact Hausdorff space X, letρX :X × T
t → Tt be defined by
ρX(x,(s,ω)) = (s,ω). Let
L0(X) ={A×B:A⊂X,Acompact,B∈ K(t)},
1RESEARCH PARTIALLY SUPPORTED BY NSF GRANT DMS-0901505
190 Electronic Communications in Probability
L1(X)the class of finite unions of sets inL0(X), andL(X)the class of intersections of countable decreasing sequences in L1(X). Let Lσ(X)be the class of unions of countable increasing se-quences of sets inL(X)andLσδ(X)the class of intersections of countable decreasing sequences of sets inLσ(X).
Lemma A.2. If A∈ B[0,t]× Ft, there exists a compact Hausdorff space X and B∈ Lσδ(X)such
that A=ρX(B).
Proof. IfA∈ K(t), we takeX = [0, 1], the unit interval with the usual topology andB=X×A. Thus the collectionM of subsets ofB[0,t]× Ft for which the lemma is satisfied containsK(t).
We will show thatM is a monotone class.
Suppose An∈ M withAn↓A. There exist compact Hausdorff spacesXn and setsBn∈ Lσδ(Xn)
such that An = ρXn(B
n). Let X =
Q∞
n=1Xn be furnished with the product topology. Let τn :
X× Tt →Xn× Tt be defined byτn(x,(s,ω)) = (xn,(s,ω))ifx= (x1,x2, . . .). LetCn=τ−n1(Bn)
and let C =∩nCn. It is easy to check thatL(X)is closed under the operations of finite unions and intersections, from which it follows that C ∈ Lσδ(X). If(s,ω)∈A, then for each n there
existsxn∈Xnsuch that(xn,(s,ω))∈Bn. Note that((x1,x2, . . .),(s,ω))∈Cand therefore(s,ω)∈ ρX(C). It is straightforward thatρX(C)⊂A, and we concludeA∈ M.
Now supposeAn∈ M withAn↑A. LetXn andBnbe as before. LetX′ =∪∞n=1Xn× {n} with the topology generated by{G× {n} :Gopen inXn}. LetX be the one point compactification of X′. We can writeBn=∩mBnmwithBnm∈ Lσ(Xn). Let
Cnm={((x,n),(s,ω))∈X× Tt:x∈Xn,(x,(s,ω))∈Bnm},
Cn=∩mCnm, andC=∪nCn. ThenCnm∈ Lσ(X)and soCn∈ Lσδ(X).
If ((x,p),(s,ω))∈ ∩m∪nCnm, then for each mthere existsnm such that ((x,p),(s,ω)) ∈Cnmm. This is only possible if nm = p for eachm. Thus((x,p),(s,ω)) ∈ ∩mCpm= Cp ⊂ C. The other
inclusion is easier and we thus obtainC=∩m∪nCnm, which impliesC∈ Lσδ(X). We check that
A=ρX(C)along the same lines, and thereforeA∈ M.
IfI0(t)is the collection of sets of the form[a,b)×C, wherea<b≤tandC∈ F
t, andI(t)is
the collection of finite unions of sets inI0(t), thenI(t)is an algebra of sets. We note thatI(t)
generates theσ-fieldB[0,t]× Ft. A set inI0(t)of the form[a,b)×Cis the union of sets in K0(t)of the form[a,b−(1/m)]×C, and it follows that every set inI(t)is the increasing union of sets inK(t). SinceM is a monotone class containing K(t), thenM containsI(t). By the monotone class theorem,M =B[0,t]× Ft.
The works of Suslin and Lusin present a different approach to the idea of representing Borel sets as projections; see, e.g.,[4, p. 88]or[3, p. 284].
Lemma A.3. If A∈ B[0,t]× Ft, then A is t-approximable.
Proof.We first prove that ifH∈ L(X), thenρX(H)∈ K
δ. IfH∈ L1(X), this is clear. Suppose that
Hn↓Hwith eachHn∈ L1(X). If(s,ω)∈ ∩nρX(H
n), there existxn∈X such that(xn,(s,ω))∈Hn.
Then there exists a subsequence such thatxnk→x∞by the compactness ofX. Now(xnk,(s,ω))∈
Hnk⊂Hmfornklarger thanm. For fixedω,{(x,s):(x,(s,ω))∈Hm}is compact, so(x∞,(s,ω))∈ Hmfor allm. This implies(x∞,(s,ω))∈H. The other inclusion is easier and therefore∩nρX(Hn) =
ρX(H). SinceρX(H
n)∈ Kδ(t), thenρX(H)∈ Kδ(t). We also observe that for fixedω,{(x,s):
Hitting times 191
Now supposeA∈ B[0,t]× Ft. Then by Lemma A.2 there exists a compact Hausdorff spaceXand B∈ Lσδ(X)such thatA=ρX(B). We can writeB=∩
nBnandBn=∪mBnmwithBn↓B,Bnm↑Bn,
andBnm∈ L(X).
Leta=P∗(π(A)) =
P∗(π◦ρX(B))and letǫ >0. By Lemma A.1,
P∗(π◦ρX(B∩B
1m))↑P∗(π◦ρX(B∩B1)) =P∗(π◦ρX(B)) =a.
Takemlarge enough so thatP∗(π◦ρX(B∩B
1m))>a−ǫ, letC1=B1m, andD1=B∩C1.
We proceed by induction. Suppose we are given sets C1, . . . ,Cn−1 and sets D1, . . . ,Dn−1 with
Dn−1=B∩(∩in=1−1Ci),P∗(π◦ρX(Dn−1))>a−ǫ, and eachCi=Bimi for somemi. SinceDn−1⊂B⊂ Bn, by Lemma A.1
P∗(π◦ρX(Dn−1∩Bnm))↑P∗(π◦ρX(Dn−1∩Bn)) =P∗(π◦ρX(Dn−1)).
We can take mlarge enough so thatP∗(π◦ρX(Dn−1∩Bnm))> a−ǫ, letCn= Bnm, and Dn=
Dn−1∩Cn.
If we letGn=C1∩ · · · ∩CnandG=∩nGn=∩nCn, then each Gn is inL(X), henceG∈ L(X).
SinceCn⊂Bn, thenG⊂ ∩nBn=B. EachGn∈ L(X)and so by the first paragraph of this proof,
for each fixedωandn,{(x,s):(x,(s,ω))∈Gn}is compact. Hence by a proof very similar to that
of Lemma 2.2,π◦ρX(G
n)↓π◦ρX(G). Using the first paragraph of this proof and Lemma 2.2, we
see that
P(π◦ρX(G)) =limP(π◦ρX(G
n))≥limP∗(π◦ρX(Dn))≥a−ǫ.
Using the first paragraph of this proof once again, we see thatAist-approximable.
Proof of Theorem 2.1. Let E be a progressively measurable set and let A= E∩([0,t]×Ω). By Lemma A.3,Aist-approximable. By Proposition 2.3,(DE ≤ t) =π(A)∈ Ft. Because t was arbitrary, we concludeDE is a stopping time.
Acknowledgement.I would like to thank the referee for valuable suggestions.
References
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