I got stuck with the following problem while following Section $10.14$ of the book 'Lectures on von Neumann algebras' by Stratila and Zsido.
Problem: Let $\mathcal{M}$ be a von Neumann algebra. Show that a normal weight $\varphi$ on $\mathcal{M}$ is lower $w$-semicontinuous, i.e. for each $\lambda \in \mathbb{R}^+, \{x\in\mathcal{M}^+:\varphi (x)\leq\lambda\}$ is $w$-closed.
Here I am providing the relevant definitions used in the previous problem.
Definition 1: Let $\mathcal{M}$ be a von Neumann algebra. A mapping $\varphi:\mathcal{M}^+\rightarrow [0,+\infty]$ is called a weight if
- $\varphi (a+b)=\varphi (a) + \varphi (b),\;a,b\in\mathcal{M}^+$,
- $\varphi(\lambda a)=\lambda\varphi (a)\;a\in\mathcal{M}^+,\lambda\geq 0$.
Definition 2: A weight $\varphi$ on a von Neumann algebra $\mathcal{M}$ is normal if there exists a family $\{\varphi_i\}$ of $w$-continuous positive forms on $\mathcal{M}$ such that $\varphi (a)=\sum_i\varphi_i(a),\;a\in\mathcal{M}^+$.
The Author says the problem is very easy to prove. Please help me with a direct proof. Thanks in advance.
The monotone increasing limit of a net of lower semicontinuous functions (on an arbitrary topological space) is lower semicontinuous. In fact, if $f_j\nearrow f$ and $\lambda>0$, then $$ f^{-1}((-\infty,\lambda])=\bigcap_j f_j^{-1}((-\infty,\lambda]), $$ which is closed as an intersection of closed sets.
In your case, $\phi$ is even a monotone increasing limit of continuous functions (namely the partial sums of the $\phi_i$).