I am trying to proof the following:
Let $(\Omega,\mu)$ be a measure space, $1<p<\infty$. Then weak convergence of $(f_n)$ to $f$ in $L^p(\Omega,\mu)$ is equivalent to $(f_n)$ being bounded and $\int_M f_n \to \int_M f$, for any measurable $M$ with finite $\mu$-volume.
I have a proven "$\Rightarrow$", by stating that weak convergence implies boundedness in general, and that integration over $M$ is a linear map to the underlying field i.e. a linear functional.
But I do not know how to approach the other direction. Of course I would be very happy to show this direction as well without making use of the isomorphism $(L^p)^* \to L^q$ for $p,q$ Hölder-conjugate, but I have not found a way yet.
(Unfortunately I don't know that much about measure theory, this is kind of a problem for me here.)