I want to find a sequence $\displaystyle f_n(x) \in L^+$ such that $\displaystyle f_n$ is defined on (0,1) for all natural numbers $\displaystyle n$, we have $\displaystyle lim_{n\rightarrow\infty}\int f_n(x)dm=0$ (m=Lebesgue measure), and $\displaystyle lim_{n\rightarrow\infty}f_n(x)$ does not exist for any x in (0,1).

My initial idea, which I know does not work, was to define something like this. For n odd, let $\displaystyle f_n(x)$ be $\displaystyle \displaystyle\frac{2}{n+1}$ for irrational x, and 0 otherwise. For n even, let $\displaystyle f_n(x)$ be 0 for irrational x and 1 for rational x. That satisfies the first condition about the limit of the integral, but does not satisfy the second condition since the limit of $\displaystyle f_n(x)$ is 0 on the irrationals.

Even though I know this doesn't work, I was wondering if someone here would know of a way I could think about tweaking this to work, or am I barking up the wrong tree, so to speak?