$ \chi ({X_{1},\dots ,X_{n}}|D) $ we take the infimum over a smaller set. The equality between $ \chi ({X_{1},\dots ,X_{n}}|\mathbb {C}) $ and $ \chi ({X_{1},\dots ,X_{n}}) $ is left to the reader.

Let $ X_{i}^{j} $, $ i=1,\dots ,n $, $ j=1,2,\dots  $ be non-commutative random variables. Assume that $ {X_{1},\dots ,X_{n}}$ are such that as $ j\to \infty  $, the joint distribution of $ (X_{1}^{j},\dots ,X_{n}^{j}) $ and $ B $ converges to the joint distribution of $ B $ and $ {X_{1},\dots ,X_{n}}$. Let $ Y_{1}^{j},\dots ,Y_{m}^{j} $ be such that they converge in distribution to $ {Y_{1},\dots ,Y_{m}}$. Assume that there exists a finite constant $ R $, so that $ \sup _{i,j}\Vert X_{i}^{j}\Vert ,\Vert Y_{i}^{j}\Vert ,\Vert X_{i}\Vert ,\Vert Y_{i}\Vert <R $. Then $$\chi ({X_{1},\dots ,X_{n}}|B)\geq \limsup _{j}\chi (X_{1}^{j},\dots ,X_{n}^{j}|B)$$ and $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\geq \limsup _{j}\chi (X_{1}^{j},\dots ,X_{n}^{j}|Y_{1}^{j},\dots ,Y_{m}^{j}).$$ The same conclusion holds for $ \chi ^{\omega } $ instead of $ \chi  $.

Clearly, only the second inequality needs to be proved. It follows from the following inclusion, true for sufficiently large $ j $: $$\begin{aligned}
\Gamma _{R}(X_{1}^{j},\dots ,X_{n}^{j}|y_{1},\dots ,y_{m}[Y_{1}^{j},\dots ,Y_{n}^{j}];k,l,\epsilon /2)\subset  &  & \\
\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}};k,l,\epsilon ), &  & \end{aligned}$$ 