Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators

Consider a von Neumann algebra $\mathcal M$ with a faithful normal semifinite trace $\tau$. We prove that each order bounded sequence of $\tau$-compact operators includes a subsequence whose arithmetic averages converge in $\tau$. We also prove a noncommutative analog of Pratt's lemma for $L_1(\mathcal M,\tau)$. The results are new even for the algebra $\mathcal{M=B(H)}$ of bounded linear operators with the canonical trace $\tau=\mathrm{tr}$ on a Hilbert space $\mathcal H$. We apply the main result to $L_p(\mathcal M,\tau)$ with $0<p\le1$ and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of $\tau$-compactness of the dominant.