Suppose I have $\lim_{n \to \infty} |f_n(x)-f(x)|=0$ for all $x$ except a set of measure $\delta$ for any $\delta>0$.
Can I say then that $\lim_{n \to \infty} |f_n(x)-f(x)|=0$ for all $x$ except a set of measure $0$.
I think this is true. We can probably prove it using contradiction. I just want to be sure.
Thanks for any help. If there is a need I can provided the actual function.
Let be $A$ the set where the $\lim\ne 0$. By hypothesis $\forall\delta>0\ m(A)<\delta$. Suppose $m(A)\ne 0$ and take $\delta=m(A)/2$.