is embeddable into the ultrapower of the hyperfinite II$ _{1} $ factor. Then $$\chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B)=\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B).$$ In particular, $$\chi (X_{1},\dots \, X_{n}|B)=\chi ({X_{1},\dots ,X_{n}}).$$ The same statements hold for $ \chi ^{\omega } $.

It is sufficient to show that $$\chi ({X_{1},\dots ,X_{n}})+\chi (X_{n+1},\dots ,X_{p}|B)\geq \chi ({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|B).$$ Fix $ m>0 $ and elements $ {Y_{1},\dots ,Y_{m}}\in B $. Let $ {y_{1},\dots ,y_{m}}\in \Gamma _{R}({Y_{1},\dots ,Y_{m}};k,l,\epsilon ) $; such an $ m $-tuple exists because $ B $ can be embedded into the ultrapower of the hyperfinite II$ _{1} $ factor. By Voiculescu’s result in [@dvv:improvedrandom], we have that the ratio $$\liminf _{k\to \infty }\frac{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon )\times \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )}{\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )}\geq 1.$$ Indeed, given a $ \delta >0 $, there is a $ k_{0} $, for all [<span style="font-variant:small-caps;">$ k>k_{0} $</span>]{} and for each choice of an approximant $$(x_{n+1},\dots ,x_{p})\in \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ),$$ there exists an open subset $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\subset \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon ),$$ so that $$\frac{\lambda \Gamma _{k,x_{n+1},\dots ,x_{p}}}{\Gamma _{R}(X_{1},\dots ,X_{n};k,l,\epsilon )}>1-\delta$$ satisfying $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\times \{(x_{n+1},\dots ,x_{p})\}\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\frac{\epsilon }{2}).$$ Let $ O=O(x_{n+1},\dots 