Ensemble average VS spatial average for a cosmological density field
I can't see the contradiction. One version says average over the regions, the other says integrate over one realisation (containing many regions). I suppose you could argue that one is performing a summation,whilst the other is an integration, but if the regions are small compared with the size of your observable universe then they amount to the same thing.
$$\frac{1}{\sum_i V_i} \sum_i \lambda_i  V_i \simeq \frac{1}{V}\int \lambda(r)\ dV$$