Operator topologies on $L^{\infty}(X,\mu )$
Solution 1:
I think you can learn something related to von Neumann algebra.
$L^2(X,\mu)$ is a Hilbert space. You can verify that $L^\infty(X,\mu)$ is a maximal abelian von Neumann algebra on $L^2(X,\mu)$.
By von Neumann bicommutant theorem (https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem), $M$ is a von Neumann algebra on Hilbert space $H$ iff $M$ is closed in strong operator topology iff $M$ is closed in weak operator topology.