  For a countably decomposable finite von Neumann algebra $\mathscr{R}$, we
show that any choice of a faithful normal tracial state on $\mathscr{R}$
engenders the same measure topology on $\mathscr{R}$ in the sense of Nelson (J.
Func. Anal., 15 (1974), 103--116). Consequently it is justified to speak of
`the' measure topology of $\mathscr{R}$. Having made this observation, we
extend the notion of measure topology to general finite von Neumann algebras
and denominate it the $\mathfrak{m}$-topology. We note that the procedure of
$\mathfrak{m}$-completion yields Murray-von Neumann algebras in a functorial
manner and provides them with an intrinsic description as unital ordered
complex topological $*$-algebras. This enables the study of abstract Murray-von
Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes
apparent the appropriate notion of Murray-von Neumann subalgebras, and the
intrinsic nature of the spectrum and point spectrum of elements, independent of
their ambient Murray-von Neumann algebra. In this context, we show the
well-definedness of the Borel function calculus for normal elements and use it
along with approximation techniques in the $\mathfrak{m}$-topology to transfer
many standard operator inequalities involving bounded self-adjoint operators to
the setting of (unbounded) self-adjoint operators in Murray-von Neumann
algebras. On the algebraic side, Murray-von Neumann algebras have been
described as the Ore localization of finite von Neumann algebras with respect
to their corresponding multiplicative subset of non-zero-divisors. Our
discussion reveals that, in addition, there are fundamental topological,
order-theoretic and analytical facets to their description.
