Pointwise convergence on dense set extension

Question

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 $X$. Suppose that $A$ is a dense subset of $X$ and $f_n$ converge point-wise to $f$ on $A$. Can we deduce that $f_n$ converge to $f$, $\mu$-a.e. on $X$?

Answer 1

Let $X=[0,1]$, $V=\mathbb{R}$, $\mu$ the Lebesgue measure on $[0,1]$, $A=\mathbb{Q}\cap [0,1]$ which is dense in $[0,1]$, and $$f_n(x)=\sin (\pi n!x) $$ Then $f_n\to 0$ on $A$, but $f_n \not \to 0$ on $[0,1]\setminus A$. Since $\mu(A)=0$, $f_n\not \to 0$ $\mu$-a.e.