Example for a sequence of functions in $\mathcal{L}^1[0,1]$

Question

I am looking for an example of a sequence of functions $(g_n)$ that is in $\mathcal{L}^1[0,1]$ and U.I. so that the following three conditions are satisfied:

$\forall \, n\, \, \vert g_n \vert \leq 1 \text{ a.e.}, \Vert g_n \Vert_1 \geq {1 \over10} \text{ and } \forall \, E \subset [0,1] \text{ and measurable}, \, \lim_{n \to \infty} \int_E g_n = 0$

Any help is appreciated.

Answer 1

It appears you're looking for a sequence of functions $g_n$ which converges weakly but not in $L^1$. A standard example is $g_n(x) = \sin (2 \pi n x)$- you should work out the properties for yourself.

One note: due to the bounded convergence theorem, if $\{g_n\}$ are all bounded from above by 1, then the sequence $\{g_n\}$ is uniformly integrable.