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I am studying the book of Reed and Simon, Functional Analysis and I am not able to prove the following exercise (16 on chapter 1):

Prove that the bounded Borel functions on $[0,1]$ are the smallest family $\mathcal{F}$ which includes $C[0,1]$ and has the property: If $f_n$ is a sequence of uniformly bounded function in $\mathcal{F}$ and $f_n\to f$ pointwise, then $f\in \mathcal{F}$.

I appreciate the tips and/or solutions.

user23069
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Hint: If you know the construction of Borel sets by transfinite induction, this shouldn't be too hard to mimic: show that every bounded Borel function can be obtained by iterating a transfinite construction of adding pointwise limits of uniformly bounded sequences.

Asaf Karagila
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  • Borel sets: smallest family of subsets with the properties: closer under complements, closer under countable unions and contains each open intervals. – user23069 Jul 12 '13 at 01:39
  • Perhaps this thread will interest you: http://math.stackexchange.com/questions/54172/the-sigma-algebra-of-subsets-of-x-generated-by-a-set-mathcala-is-the-s/ – Asaf Karagila Jul 12 '13 at 01:42