TITLE: How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure?
QUESTION [6 upvotes]: It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to $\sigma$-finite measures. What about arbitrary measures. Is $L_1(\mu)$ always weakly sequentially complete?

REPLY [5 votes]: The support of any $L_1$ function is $\sigma$-finite.  A countable union of $\sigma$-finite sets is $\sigma$-finite.  So a sequence of $L_1$ functions is supported by a $\sigma$-finite set.  Now you are in the "easy extension" mentioned.