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I'm thinking about the following exercise from Intro to stoch. analysis:

Let [itex]V[/itex] be a continuous, nondecreasing function on [itex]\mathbb{R}[/itex] and [itex]\Lambda[/itex] its Lebesgue-Stieltjes measure. Say [itex]t[/itex] is a point of strict increase for [itex]V[/itex] if [itex] V(s) < V(t) < V(u)[/itex] for all [itex]s<t[/itex] and all [itex]u>t[/itex]. Let [itex]I[/itex] be the set of such points. Show that [itex]I[/itex] is a Borel set and [itex]\Lambda(I^C) = 0[/itex].

My attempt to this exercise:

By definition, for any rational point [itex]t \in I[/itex] there exists an [itex]\epsilon[/itex]-neighbourhood of [itex]t[/itex] containing [itex]s<t[/itex] and [itex]u>t[/itex] where [itex]s,u \in I[/itex]. The neighbourhood, forming an open set, is Borel. Countable (due to rationality of [itex]t[/itex]s) union of these neighbourhoods forms [itex]I[/itex] which is Borel too.

Complement of [itex]I[/itex] is hence a countable union of connected sets (say [itex]J_i, i=1,...,n[/itex]) where, due to the nondecreasing property [itex]V(x) = V(y)[/itex] for all [itex]x,y[/itex] in particular [itex]J_i[/itex]. Since [itex]\Lambda(J_i) = |V(x)-V(y)| = 0[/itex] for any connected set [itex]J_i[/itex], hence [itex]\Lambda(I^C) = \Lambda(\cup J_i) = 0[/itex].

Intuitively, I feel that my proof misses or skips something important...