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Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
Suppose that $A$ is a dense subset of $X$ and $f_n$ converge point-wise to $f$ on $A$.

Does there exist a subset $K$ of $X$, $\mu$-measure epsilon such that $f_n$ converge to $f$, $\mu$-a.e. on $X$?

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  • $\begingroup$ No, not in general, try $\mu=\delta_x$. $\endgroup$
    – Christian Remling
    Commented Aug 19, 2018 at 17:45
  • $\begingroup$ What if $\mu$ has no atoms and is $\sigma$-finite? $\endgroup$
    – ABIM
    Commented Aug 19, 2018 at 18:22
  • $\begingroup$ Not sure, consider $sin(xn!\pi)$ $\endgroup$
    – ABIM
    Commented Aug 19, 2018 at 18:22

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