  Let $\M$ be a type ${\rm II_1}$ factor and let $\tau$ be the faithful normal
tracial state on $\M$. In this paper, we prove that given finite elements
$X_1,\cdots X_n \in \M$, there is a finite decomposition of the identity into
$N \in \NNN$ mutually orthogonal nonzero projections $E_j\in\M$,
$I=\sum_{j=1}^NE_j$, such that $E_jX_iE_j=\tau(X_i) E_j$ for all $j=1,\cdots,N$
and $i=1,\cdots,n$. Equivalently, there is a unitary operator $U \in \M$ such
that $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jX_iU^j=\tau(X_i)I$ for $i=1,\cdots,n$.
This result is a stronger version of Dixmier's averaging theorem for type ${\rm
II}_1$ factors. As the first application, we show that all elements of trace
zero in a type ${\rm II}_1$ factor are single commutators and any self-adjoint
elements of trace zero are single self-commutators. This result answers
affirmatively Question 1.1 in [10]. As the second application, we prove that
any self-adjoint element in a type ${\rm II}_1$ factor can be written a linear
combination of 4 projections. This result answers affirmatively Question 6(2)
in [15]. As the third application, we show that if $(\mathcal{M},\tau)$ is a
finite factor, $X \in \mathcal{M}$, then there exists a normal operator $N \in
\mathcal{M}$ and a nilpotent operator $K$ such that $X= N+ K$. This result
answers affirmatively Question 1.1 in [9].
