Image of a bounded sequence

Question

If $(u_n)$ is bounded in $L^1$ can we say that $F(u_n)$ is bounded in $L^1$ where $F(t)=\int_0^t f(s) ds$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ is of class $C^1$

Answer 1

In general $F\left(u_n\right)$ may not be integrable. For example, we can take $f(s):=s$ (which is $C^1$). We get $F(t)=t^2/2$ hence any sequence $u_n$ such that one of the $u_n$'s is not integrable will give a counter-example. For example, take $u_n=u$ where $u$ is an integrable function but not square integrable. In this case, the sequence $(u_n)_n$ is bounded in $L^1$ but $(F(u_n))$ is not, since $F(u_n)=F(u)=u^2/2$ is not integrable.