On an analog of the M. G. Krein theorem for measurable operators

Let ${\mathcal M}$ be a von Neumann algebra of operators on a Hilbert space $\mathcal H$ and $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$. Let $\mu_t(T)$, $t>0$, be a rearrangement of a $\tau$-measurable operator $T$. Let us consider a $\tau$-measurable operator $A$, such that $\mu_t(A)>0$ for all $t>0$ and assume that $\mu_{2t}(A)/\mu_t(A) \to 1$ as $t \to \infty$. Let a $\tau$-compact operator $S$ be so that the operator $I+S$ is right invertible, where $I$ is the unit of ${\mathcal M}$. Then, for a $\tau$-measurable operator $B$, such that $A=B(I+S)$, we have $\mu_{t}(A)/\mu_t(B) \to 1$ as $t \to \infty$. It is an analog of the M.G. Krein theorem (for $\mathcal{M}=\mathcal{B}(\mathcal{H})$ and $\tau =\mathrm{tr}$, theorem 11.4, ch. V [Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs. Vol. 18. Providence, R.I., Amer. Math. Soc., 1969. 378 p.] for $\tau$-measurable operators.