  We prove that any separable II$_1$ factor $M$ admits a {\it coarse
decomposition} over the hyperfinite II$_1$ factor $R$, i.e., there exists an
embedding $R\hookrightarrow M$ such that $L^2M\ominus L^2R$ is a multiple of
the coarse Hilbert $R$-bimodule $L^2R \overline{\otimes} L^2R^{op}$
(equivalently, the von Neumann algebra generated by left and right
multiplication by $R$ on $L^2M\ominus L^2R$ is isomorphic to
$R\overline{\otimes}R^{op}$). Moreover, if $Q\subset M$ is an infinite index
irreducible subfactor, then $R\hookrightarrow M$ can be constructed so that to
also be coarse with respect to $Q$. This result implies existence of MASAs that
are mixing, strongly malnormal, and with infinite multiplicity, in any
separable II$_1$ factor.
