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I have a sequence $(f_n)_n : [0,1] \to [0,1]$ of strictly increasing functions.

$f_n$s are, in general, not left or right continuous. If they were right continuous, from what I know, we have results that show that there exists a function $f$ such that $f_n \to f$ in the weak* topology.

Are there any results that suggest that this sequence $f_n$ converges to $f$ in some sense? Would it help if I knew that $f_n$s are usc?

avk255
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  • The notion of weak-* convergence involves topological vector space. Which space are you considering here? – Sangchul Lee May 05 '18 at 08:50
  • The space is a space of weakly increasing functions that map $[0,1]$ to $[0,1]$. – avk255 May 05 '18 at 09:15
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    Helly's selection theorem? – user539887 May 05 '18 at 09:53
  • Correct of I am wrong, but except that identifying such functions with Lebesgue-Stieltjes measures on $[0,1]$ and then embedding them into the dual space of $C([0,1])$ (which essentially induces the vague topology on them), I see no clear way of equipping a topological vector space structure (and hence the weak-* topology) on the family of weakly increasing functions $[0,1]\to[0,1]$... – Sangchul Lee May 05 '18 at 10:10

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