Determine if a sequence of functions is uniformly integrable

Question

Let {$f_n$} be a sequence of continuous functions on $[0,1]$, and $|f_n|\le1$ on $[0,1]$. Is {$f_n$} uniformly integrable over $[0,1]$?

For the same sequence, if we assume {$f_n$} is integrable and $\int_0^1|f_n|\le 1$, then is {$f_n$} uniformly integrable over $[0,1]$?

Answer 1

The answer to the first question is positive. In general if $(f_n)$ is dominated by some function $g$ a.e., i.e. $|f_n|\le g$, where $g\in L^1(\Omega)$, then $(f_n)$ is uniformly integrable. Here we can take $g \equiv 1$ and $\Omega = [0, 1]$.

The answer to the second statement is No. A classic counter-example is $f_n(x) = 2n^2x\mathbf{1}_{[0, 1/2n]}-2n^2(x-1/n)\mathbf{1}_{(1/2n, 1/n]}$.

Answer 2

For the second question the classic example is $f_n(x)=(n+1)x^n.$ Here $\int_0^1 f_n =1$ for all $n,$ while for any $\delta>0,$ $\int_{1-\delta}^1 f_n = 1-(1-\delta)^{n+1}\to 1.$