Corollary 40.8.4.label Let $(X, \cm, \mu)$ be a localisable measure space, then $\Omega(L^{\infty}(X))$ is extremally disconnected.
Proof. By Theorem 40.8.3, $L^{\infty}(X; \real)$ is order complete. By Corollary 38.6.4, $\Omega(L^{\infty}(X))$ is extremally disconnected.$\square$
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