A sequence of functions $f_n : [0, 1] \to \mathbb{R}$ which converges to 0 in $L^1$, or in other words $\int |f_n| \to 0$, but doesn't converge anywhere pointwise. I will offer hints (the canonical example is kinda cool).

Question

A sequence of functions $f_n : [0, 1] \to \mathbb{R}$ which converges to 0 in $L^1$, or in other words $\int |f_n| \to 0$, but doesn't converge anywhere pointwise.  I will offer hints (the canonical example is kinda cool).

anonymous · Answer

Could be a constant sequence of a single function that meets the requirements.

anonymous · Answer

A constant sequence would certainly converge pointwise.

anonymous · Answer

If you need me to define any of the terms, I will be happy to.

anonymous · Answer

The second condition requires that for all $x$ we have that $\lim_{n\to \infty}f_n(x)$ does not exist.