Question: Is there a published research paper which already solves the problems in the summary of my paper?

Since the writing in the paper is difficult, here is the summary:

Let $n\in\mathbb{N}$ and suppose $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ is a function, where $A$ and $f$ are Borel. We want a unique, satisfying average of highly discontinuous $f$, taking finite values only. For instance, consider an everywhere surjective $f$, where its graph has zero Hausdorff measure in its dimension and 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. Thus, take any chosen family of bounded functions converging to $f$ with the same satisfying and finite expected value, where the term "satisfying" is explained in the third paragraph.

The importance of this solution is that it solves the following problem: the set of all $f\in\mathbb{R}^{A}$ with a finite expected value, forms a shy "measure zero" subset of $\mathbb{R}^{A}$. This issue is solved since the set of all $f\in\mathbb{R}^{A}$, where there exists a family of bounded functions converging to $f$ with a finite expected value, forms a prevalent "full measure" subset of $\mathbb{R}^{A}$. Despite this, the set of all $f\in\mathbb{R}^{A}$—where two or more families of bounded functions converging to $f$ have different expected values—forms a prevalent subset of $\mathbb{R}^{A}$. Hence, we need a choice function which chooses a subset of all families of bounded functions converging to $f$ with the same satisfying and finite expected value.

Notice, "satisfying" is explained in a leading question which uses rigorous versions of phrases in the former paragraph and the "measure" of the chosen families of each bounded function's graph involving partitioning each graph into equal measure sets and taking the following—a sample point from each partition, pathways of line segments between sample points, lengths of line segments in each pathway, removed lengths which are outliers, remaining lengths which are converted into a probability distribution, and the entropy of the distribution. In addition, we define a fixed rate of expansion versus the actual rate of expansion of a family of each bounded function's graph.

(Keywords: Discontinuity, Hausdorff measure, Expected Value, Function Space, Prevalent and Shy Sets, Partitions, Samples, Euclidean Distance, Entropy, Choice Function)

I used Reserachgate to find similar paper based on the keywords, but since I am an undergraduate I do not fully understand the papers. (I cannot give an accurate summary.)

Here are the most similar papers I could find.

1. Ergodic Optimization For Open Expanding Multi-Valued Topological Dynamical Systems
2. Typical Uniqueness in Ergodic Optimization
3. Prediction of dynamical systems from time-delayed measurements with self-intersections
4. A Hausdorff-measure boundary element method for acoustic scattering by fractal screens

Question 2: Do any of these papers solve the problems in the summary?

posted 11 months ago

CC BY-SA 4.0

11mo ago

bharathk98‭

-2 reputation 15 3 -17 190