Let $(X,\Sigma,\mu)$ be a finite measure space and $E$ be a separable Banach space (not $0$-dimensional). Then, what are some examples of non-linear functionals on the Bochner-space $L^1_{\mu}(X,E)$ of $E$-valued weakly-measurable functions satisfying $\int_{x \in X} \|f(x)\|_E d\mu(x)<\infty$?
In particular, when does the functional $f\mapsto \int_{x \in X} F(f(x))d\mu(x)$ work for some measurable function $F$?