**Problem:** Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\tau$ on $\widetilde{M}_+$ by $\tau(a):=\int_0^\infty\lambda\tau(e_\lambda)$ where $a=\int_0^\infty\lambda\,de_\lambda$ is the spectral decomposition (See Equation 4.6 of Hiwi). If $\{x_n\}_{n\in\mathbb{N}}$ be a sequence of positive elements from $M$, which converges strongly to an element $x$ of $\widetilde{M}_+$, then can we say that $\{\tau(x_n)\}_{n\in\mathbb{N}}$ converges to $\tau (x)$?

Or, can we at least say that if $\{x_n\}_{n\in\mathbb{N}}$ be a sequence of positive elements from $M$, which converges strongly to an element $x$ of $\widetilde{M}_+$, then $\{x_n\}_{n\in\mathbb{N}}$ converges to $x$ in the measure topology?

**(A positive answer of any one of the above two questions would be sufficient for me.)**

I got stuck with this problem while reading $\tau$-measurable operators from the book 'Lectures on Selected Topics in von Neumann Algebras' by Hiwi. Here I recall the definition of $\tau$-measurable operator.

**Definition 1:** For each $\epsilon,\delta>0$, define $$\mathscr{O}(\epsilon,\delta)=\{m\text{ affiliated to } M:eH\subseteq \mathcal{D}(m),\,\|me\|\leq \epsilon \text{ and }\tau(1-e)\leq\delta \text{ for some } e\in Proj(M)\}.$$
Let $m$ be a densely defined closed operator such that $m$ is affiliated to $M$. We say that $m$ is $\tau$-measurable if for any $\delta >0$, there exists an $\epsilon >0$ such that $m\in\mathscr{O}(\epsilon,\delta)$. We denote by $\widetilde{M}$ the set of such $\tau$-measurable operators.

**Theorem 2:** (Theorem 4.12 of Hiwi) The $\widetilde{M}$ is a complete metrizable Hausdorff topological *-algebra with $\{\widetilde{M}\cap\mathscr{O}(\epsilon,\delta):\epsilon,\delta >0\}$ as a neighborhood basis of $0$. Moreover, $M$ is dense in $\widetilde{M}$.

Thanks in advance for any help or suggestion.