# Dense Egoroff theorem

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 $X$?

• No, not in general, try $\mu=\delta_x$. – Christian Remling Aug 19 '18 at 17:45
• What if $\mu$ has no atoms and is $\sigma$-finite? – N00ber Aug 19 '18 at 18:22
• Not sure, consider $sin(xn!\pi)$ – AIM_BLB Aug 19 '18 at 18:22