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Let $X \subset \mathbb{R}$ be a compact, and let $f: X \to \mathbb{C}$ be a bounded Borel function.

Is it true that there is a sequence $\{f_n\} \subset C(X)$ s.t for any regular Borel measure $\int f_n d\mu \to \int f d\mu$?

If so, can you provide a reference, or a proof? Is it possible to show this with tools no more advanced than the Riesz Representation theorem?

I need this to complete my proof suggested here:

Spectral Theorem - $AB = BA \implies B\Phi(f) = \Phi(f)B$

Mariah
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  • There is a NET s.t. $\int f_\lambda d\mu\to \int f d\mu \forall $ regular Borel measure $\mu$, not sequence. – C. Ding Oct 21 '17 at 07:57
  • @C.Ding I see. If you'd like, check out the linked post in the question. I'm looking for an elementary proof that those operators commute. I think the proof I gave there is correct with your statement (but unfortunately I don't have the tools to formalise a proof of the existence of this net). Maybe you'll have an idea. – Mariah Oct 21 '17 at 08:33
  • @C.Ding will appreciate if you have some input for my new answer here: https://math.stackexchange.com/questions/2466162/spectral-theorem-ab-ba-implies-b-phif-phifb – Mariah Oct 21 '17 at 10:53

1 Answers1

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There is no such sequence in general. By taking Dirac delta measures we see that $f_n\to f$ pointwise everywhere. So $f$ must be of Baire class 1. But the Dirichlet function $1_{[0,1]\cap \mathbb Q}$ is not of Baire class 1 because it has no points of continuity.

Dap
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