If it’s a Bochner integral why does Takesaki justify its existence with a Riemann-Stieltjes sum? I think you can get away by skipping the justification entirely.
My original phrasing was bad in the ”there exists continuous seminorms”. Since these are precisely the seminorms that bound the product, you should state that in the beginning.
You probably don’t need $* \in \{\cap, \cup\}$ if you only use it for $\cap$. The union for (O2) shouldn’t be over the product $I_{j} \times J$, since $I_{j}$ depends on $J$. Instead, it should be something like: Let $\{U_{j}\}_{j \in J}\subset \tau(\mathcal{B})$. For each $j \in J$, let $I_{j}$ such that $U_{j} = \bigcup_{i \in I_j}B_{i, j}$, then
You probably don’t need $* \in \{\cap, \cup\}$ if you only use it for $\cap$. The union for (O2) shouldn’t be over the product $I_{j} \times J$, since $I_{j}$ depends on $J$. Instead, it should be something like: Let $\{U_{j}\}_{j \in J}\subset \tau(\mathcal{B})$. For each $j \in J$, let $I_{j}$ such that $U_{j} = \bigcup_{i \in I_j}B_{i, j}$, then