It is known that when $\mu$ is $\sigma$-finte measure, then $L^\infty(\mu)$ is $1$-injective.
But I want to know whether it is right for any $L^\infty$ spaces.



When $L^\infty$ is 1-injective
A useful sufficient condition is that $(X,\Sigma,\mu)$ is a [localizable](http://ncatlab.org/nlab/show/localizable%20measure) measure space. See 363R of [Fremlin](http://www.essex.ac.uk/maths/people/fremlin/mt.htm), Volume 3, p.308.


