By Peskir G.

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**Additional resources for A Change-of-Variable Formula with Local Time on Curves**

**Sample text**

As ε ↓ 0 over a subsequence. 48) 0 it follows that µ1,2 (Ac ∩ B c ) = 0 outside a P -null set. s. 43). 28). 19). 5. 26). 4) as claimed. 49) 0 for ε > 0. 28) is satisﬁed for all δ > 0. 28) as claimed. 31) holds then s → F (s, b(s) ± ε) is decreasing on [0, t] and therefore of bounded variation. 28) follows as well. 28) as claimed. 28) follows in the same way. 28) as claimed. 6. 49) above. 50) 0<ε<δ 0 where the ﬁnal (strict) inequality follows from the fact that F and Fx are locally bounded on C and D.

28) as claimed. 31) holds then s → F (s, b(s) ± ε) is decreasing on [0, t] and therefore of bounded variation. 28) follows as well. 28) as claimed. 28) follows in the same way. 28) as claimed. 6. 49) above. 50) 0<ε<δ 0 where the ﬁnal (strict) inequality follows from the fact that F and Fx are locally bounded on C and D. 28) is satisﬁed for all δ > 0. 27) holds. 27), respectively. 51) 0 for ε > 0. 52) where the ﬁnal (strict) inequality follows from the fact that F, LX F, µFx and Fx are locally bounded on C and D.

B(s) + εn } for n 1 and note that An ↑ A Set An = {0 s t | Xs as n → ∞, where A = {0 s t | Xs > b(s)}. 44) 0 for all n 1. This shows that µ1,2 (∂An ) = 0 for all n 1 outside a P -null set. In exactly the same way one ﬁnds that µ1,2 (∂A) = 0 outside a P -null set. ± ± Denoting µ± n =µεn for all n 1 we claim that limn→∞ µn (An )=µ1,2 (A) ± outside a P -null set. To see this set anm = µn (Am ) and note that the two limits an∞ = limm→∞ anm = µ± n (A) and a∞m = limn→∞ anm = µ1,2 (Am ) exist (the latter outside a P -null set by the weak convergence established) and moreover satisfy limn→∞ an∞ = limm→∞ a∞m = µ1,2 (A) =: a∞∞ (the former outside a P -null set by the weak convergence established).