Density of bounded functions in $L^1(0,T;L^1(M))$?

Question

Let $u \in L^1(0,T;L^1(M))$ where $M$ is a compact Riemannian manifold. Is it possible to find $u_n$ such that $u_n \to u$ in $L^1(0,T;L^1(M))$ and $u_n$ are bounded everywhere or almost everywhere on $(0,T)\times M$?

I think: yes, since $L^1(0,T;L^1(M)) = L^1((0,T)\times M)$ and $C^\infty((0,T)\times M)$ is dense in that space. Well at least this is true for bounded domains $M=\Omega$..

Answer 1

Yes. You can approximate $u$ via $u_k = \max(-k,\min(u,k))$.