  In [10], Halmos proved an interesting result that the set of irreducible
operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt
approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an
operator $a\in \mathcal M$ is said to be {irreducible in} $\mathcal M$ if
$W^*(a)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(a)'\cap
\mathcal M={\mathbb C} \cdot I$. In this paper, let $\Phi(\cdot)$ be a
$\Vert\cdot\Vert$-dominating, unitarily invariant norm (see Definition 2.1),
where by $\Vert\cdot\Vert$ we denote the operator norm. We prove that in every
semifinite von Neumann factor $\mathcal M$ with separable predual, if the norm
$\Phi(\cdot)$ satisfies a natural restriction introduced in (1.1), then
irreducible operators are $\Phi(\cdot)$-norm dense in $\mathcal M$. In
particular, the operator norm $\Vert\cdot\Vert$ and the $\max\{\Vert\cdot\Vert,
\Vert\cdot\Vert_p\}$-norm (for each $p>1$) naturally satisfy the condition in
(1.1), where $\tau$ is a faithful, normal, semifinite, tracial weight and
$\Vert x\Vert_p=\tau(|x|^p)^{1/p}$ for all $x\in \mathcal M \cap L^{p}(\mathcal
M,\tau)$ (see [18, Preliminaries]). This can be viewed as a (stronger) analogue
of a theorem of Halmos in [10], proved with different techniques developed in
semifinite, properly infinite von Neumann factors.
  Meanwhile, for every $\Vert\cdot\Vert$-dominating, unitarily invariant norm
$\Phi(\cdot)$, we develop another method to prove that each normal operator in
$\mathcal M$ is a sum of an irreducible operator in $\mathcal M$ and an
arbitrarily small $\Phi(\cdot)$-norm perturbation, where the $\Phi(\cdot)$-norm
isn't restricted by (1.1). Particularly, the $\Phi(\cdot)$-norm can be the
$\max\{\Vert\cdot\Vert, \Vert\cdot\Vert_1\}$-norm.
