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#4: Post edited
by
bharathk98
·
2025-10-14T15:02:38Z (11 months ago)
Papers which solve the summary of my paper involving expectations, Hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
- Papers which solve the summary of my paper involving expectations, hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
#3: Post edited
by
bharathk98
·
2025-10-14T15:01:59Z (11 months ago)
Papers which solve the summary of my paper involving expectations, hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
- Papers which solve the summary of my paper involving expectations, Hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
#2: Post edited
by
bharathk98
·
2025-10-14T14:59:49Z (11 months ago)
Made edits using @Tommis suggestions in the comments: changed title and added attempt to find a similar paper.
Is there a research paper which already solves the problem presented in the summary of my paper?
- Papers which solve the summary of my paper involving expectations, hausdorff measure, prevelant/shy sets, partitions, samples, pathways & entropy.
- **Question:** Is there a published research paper which already solves the problems in the summary of [my paper][1]?
- 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][2]
- > "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][2] "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][2] 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][3] 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.
- [1]: https://www.researchgate.net/publication/396439221_Averaging_Highly_Discontinuous_Functions_With_Undefined_Expected_Values_Using_Families_Of_Bounded_Functions
- [2]: https://en.wikipedia.org/wiki/Prevalent_and_shy_sets
[3]: https://en.wikipedia.org/wiki/Entropy_(information_theory)
- **Question:** Is there a published research paper which already solves the problems in the summary of [my paper][1]?
- 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][2]
- > "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][2] "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][2] 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][3] 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][4]
- 2. [Typical Uniqueness in Ergodic Optimization][5]
- 3. [Prediction of dynamical systems from time-delayed measurements with self-intersections][6]
- 4. [A Hausdorff-measure boundary element method for acoustic scattering by fractal screens][7]
- **Question 2:** Do any of these papers solve the problems in the summary?
- [1]: https://www.researchgate.net/publication/396439221_Averaging_Highly_Discontinuous_Functions_With_Undefined_Expected_Values_Using_Families_Of_Bounded_Functions
- [2]: https://en.wikipedia.org/wiki/Prevalent_and_shy_sets
- [3]: https://en.wikipedia.org/wiki/Entropy_(information_theory)
- [4]: https://arxiv.org/pdf/2503.18092
- [5]: https://arxiv.org/pdf/2506.01518
- [6]: https://www.sciencedirect.com/science/article/pii/S0021782424000345
- [7]: https://link.springer.com/article/10.1007/s00211-024-01399-7
#1: Initial revision
by
bharathk98
·
2025-10-13T23:30:28Z (11 months ago)
Is there a research paper which already solves the problem presented in the summary of my paper?
**Question:** Is there a published research paper which already solves the problems in the summary of [my paper][1]?
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][2]
> "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][2] "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][2] 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][3] 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.
[1]: https://www.researchgate.net/publication/396439221_Averaging_Highly_Discontinuous_Functions_With_Undefined_Expected_Values_Using_Families_Of_Bounded_Functions
[2]: https://en.wikipedia.org/wiki/Prevalent_and_shy_sets
[3]: https://en.wikipedia.org/wiki/Entropy_(information_theory)