Defining a Satisfying Expected Value From Chosen Sequences of Bounded Functions Converging to Pathological Functions

Let n ∈ N and suppose function f : A ⊆ R^n → R, where A and f are Borel. We want a satisfying average for all pathological f (e.g., everywhere surjective f whose graph has zero Hausdorff measure in its dimension) taking finite values only. If this is impossible, we wish to average a nowhere continuous f defined on the rationals. The problem is that the expected value of these examples of f , w.r.t the Hausdorff measure in its dimension, is undefined. We fix this by taking the expected value of chosen sequences of bounded functions converging to f with the same satisfying and finite expected value. Note, “satisfying” is explained in the leading question which uses rigorous versions of phrases in the former paragraph and the “measure” of a bounded functions’ graph which involves minimal pair-wise disjoint covers of the graph with equal ε measure, sample points from each cover, paths of line segments between sample points, the lengths of the line segments in the path, removed lengths which are outliers, remaining lengths which are converted into a probability distribution, and the entropy of the distribution. We also explain “satisfying” by defining the actual rate expansion of a bounded functions’ graph and also “the rate of divergence” of a bounded functions’ graph compared to that of other bounded functions’ graphs.