Finding a Research Paper Which Meaningfully Averages Unbounded Sets

Suppose n ∈ N. We wish to meaningfully average ‘sophisticated’ unbounded sets (i.e., sets with positive n-d Hausdorff measure, in any n-d box of the n-d plane, where the measures do not equal the area of the boxes). We do this by taking an unique, satisfying extension of the expected value, w.r.t the Hausdorff measure in its dimension, on bounded sets taking finite values only. As of now, I’m unable to define this due to limited knowledge of advanced math and most people are too busy to help. Therefore, I’m wondering if anyone knows a research paper which solves my doubts. (Unlike previous versions with similar names, we made everything rigorous.)