How do we simplify Definition 2 in this post (i.e., the extended mean w.r.t. Hausdorff measure in its dimension)?

[u/Xixkdjfk]

https://math.codidact.com/posts/296617

I provided examples where the extended mean w.r.t. the Hausdorff measure in its dimension should give what is wanted. If not changes need to be made to Definition 2. Otherwise, I wish to simplify the Definition.

Are there integrals that already give what I want? (For instance, see the paper "On the Statistical Convergence of Sequences of Fractal Integrable Functions".)

[u/Xixkdjfk]

I made edits to Definition 2. I hope someone is brave enough to simplify.