Representation of Quasi-Measure by Henstock–Kurzweil Type Integral on a Compact-Zero Dimensional Metric Space

Скворцов, В. А.; Tulone, Francesco

doi:10.1515/gmj.2009.575

“…Hence, 3. In the case f is unbounded and fractal, we could use [10] (p.19-47), which applies a Henstock-Kurzweil type integral (i.e., µ-HK integral) on a measure Metric Space. This coincides with unbounded functions with finite improper Riemman integrals, including bounded functions with finite Lebesgue integrals, bounded function with finite integrals w.r.t the Hausdorff measure, or function with finite Henstock-Kurzweil integrals.…”

Section: Extended Expected Valuesmentioning

confidence: 99%

“…Further, we assume using Section 1.2, crit. 1, there is no (exact) dimension function of A nor could A be "fractal" enough for extensions of the Lebesgue Density Theorem [8], extensions of the Hausdorff measure using Hyperbolic Cantor Sets [9], or extension of the Henstock-Kurzweil integral on the Metric Space [10] (p.19-47).…”

Section: Examplesmentioning

confidence: 99%

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Krishnan

2024

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Suppose for n∈ ℕ, set A&sube;ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; we want to find an unique and “natural” extension of the expected value, w.r.t the Hausdorff measure, that's finite for all f in a non-shy subset of ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of ℝᴬ. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x₁,...,xₙ,fₖ*(x₁,...,xₙ))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, if (Fₖ***) converges to A at a rate linear or super-linear to the rate non-equivelant sequences of sets converge, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique and “natural” extension (defined on §2.3 & §2.4) for all f in a non-shy subset of ℝᴬ. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.

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“…another example is: (11) Note this leads to a new extension of the expected value where when there's at least one starred-sequence of sets where the extension is finite, the extension could be finite for all f in a non-shy subset of all Borel measurable functions in R A .…”

Section: Attempt To Answer Thesismentioning

confidence: 99%

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Krishnan

2023

Preprint

0

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Suppose for n∈ ℕ, set A&sube;ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; we want to find an unique and “natural” extension of the expected value, w.r.t the Hausdorff measure, that's finite for all f in a non-shy subset of ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of ℝᴬ. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x₁,...,xₙ,fₖ*(x₁,...,xₙ))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, if (Fₖ***) converges to A at a rate linear or super-linear to the rate non-equivelant sequences of sets converge, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique and “natural” extension (defined on §2.3 & §2.4) for all f in a non-shy subset of ℝᴬ. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.

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Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space

Tulone¹

2011

Tatra Mountains Mathematical Publications

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Representation of Quasi-Measure by Henstock–Kurzweil Type Integral on a Compact-Zero Dimensional Metric Space

Abstract: A derivation basis is introduced in a compact zero-dimensional metric space 𝑋. A Henstock–Kurzweil type integral with respect to this basis is defined and used to represent the so-called quasi-measure on 𝑋.

Cited by 4 publications

References 5 publications

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions

Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space

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