Suppose that $u\_k$ is a sequence of $L^1$ functions defined on a compact $K\subset R^n$ and a function $f:[0, \infty)\to[0, \infty)$ with the following properties

* $u\_k\ge 0$
* $\|u\_k\|\_{L^1}=\int u\_k=1$
* $u\_k\to u$ strongly in $L^1$
* $f$ is convex, $f(0)=0$ and has superlineair growth at $+\infty$ (that is: $\lim\_{z\to+\infty} \frac{f(z)}{z}=+\infty$)
* $\int f(u\_k)< C$ for all $k$

Note that the conditions on $f$ imply that the functional $v\mapsto\int f(v)$ is lower semicontinuous with respect to weak $L^1$ convergence.

Does this imply that $f(u\_k)$ converges to $f(u)$ weakly in $L^1$, possibly under the extra assumption that $\int f(u\_k)\to \int f(u)$?

**EDIT**: Thanks for the answers, both were helpful and received an upvote.