I highly recommend Segal's original paper *Equivalences of Measure Spaces*
[American Journal of Mathematics
Vol. **73**, No. 2 (1951), pp. 275-313], where he introduced localizable spaces, since this was before the terminology took off.
In it he shows that an arbitrary measure space has maximal abelian (i.e. strongly closed) $L^\infty$ algebra if and only if it is localizable.
So there do exist measure spaces which for which $L^\infty(X)$ is not a von Neumann algebra. But (and it's a big but), if you're at all interested in integration, then the class of localizable measure spaces is really as large a class of interesting measure spaces as there is. The following excerpt explains why:

*The class of measure spaces with these properties (we call such spaces "localizable") constitutes in some ways a more natural generalization of the $\sigma$-finite measure spaces, than the class of arbitrary measure spaces. In particular, for a measure space to be localizable is equivalent to the validity for the space of the conclusion of the Radon-Nikodym theorem, or alternatively to the conclusion of the Riesz representation theorem for continuous linear functionals on the Banach space of integrable functions. Every measure space is metrically equivalent (by which we mean there is a measure-preserving isomorphism between the $\sigma$-finite measure rings - roughly speaking this means the spaces are equivalent as far as integration over them is concerned) to a localizable space, and this latter space is essentially unique.*

The point of view being held here is that measure theory is not about defining odd pathological measure spaces, but about all the cool stuff you can build on the nicer spaces: integration theory, probability theory, dynamical systems, stochastic processes, ergodic theory. And all that interesting stuff can be done, or is being worked out, in von Neumann algebras.

So von Neumann algebras capture the integration bit of measure theory.

Regarding your point in the comments: "**since in vNT all what distinguishes measure theory from topology (mainly measurable spaces not being topological) is absent (we are restricted to LMS, which are topological)?**" The category of localizable measure spaces and measurable maps is equivalent to the category of hyperstonean topological spaces and hyperstonean maps - this is not a subcategory of topological spaces and continuous maps - the maps between these spaces have to preserve an extra structure, namely a family of normal measures.
(Just like the category of topological spaces is not a subcategory of sets, because again they have different morphisms, but there is a forgetful functor Top-->Set).

hyperstonean/hyperstonianthen. One can even have `a formula' for that compact space: $K=\mbox{Stone space}(\wp(\Omega)/\mathcal{N}_\mu)$, where $(\Omega, \mathcal{F}, \mu)$ is your measure space and $\mathcal{N}_\mu$ denotes the σ-ideal of $\mu$-null sets. Please have a look on vol. 1 of Takesaki's book for more details. – Tomek Kania Jul 24 '13 at 13:10interestingnon-localisable measure spaces though. – Tomek Kania Jul 24 '13 at 13:39