Graphs of real functions with pathological behaviors

Bernardi, Claudio

doi:10.1007/s00500-016-2342-4

Cited by 1 publication

(3 citation statements)

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“…Note, everywhere, surjective functions, e.g. f , have been well studied [2][3][4]; however, their expected values (w.r.t the Lebesgue or Hausdorff measure) have not. Moreover, a meaningful average of f (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…the main question of §3. 4) where the answer to the focus [3] is E * * [ f , F ⋆⋆⋆ k ] in eq. 48-using def.…”

mentioning

confidence: 99%

“…47 that's undefined due to division by infinity.) [3] If the set B * * ⊆ B * (def. 7) is the set of all f ∈ B * , with an unique and "meaningful" extension of the expected value-w.r.t the Hausdorff measure-on bounded functions to bounded/unbounded f taking finite values, then B * * should be non-shy subset of B *…”

mentioning

confidence: 99%

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Meaningfully Averaging Unbounded Functions

Krishnan

2024

Preprint

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In this paper we want meanignfully average an "infinite collection of objects covering an infinite expanse of space". For any n∈ ℕ, set A&sube;ℝⁿ and set B&sube;ℝ where (A,P) is a Polish space, we illustrate this quote with an everywhere, surjective function f∶A→B. The problem is no meaningful expected value of f (e.g., w.r.t the Lebesgue or Hausodorff measure) on Borel sets has a finite value, since the graph of f in any n-dim. interval which covers a subset of A×B has countably infinite points. (The Hausdorff measure of countably infinite points is +∞, where the expected value of f is undefined due to division by infinity.) To fix this, we need the most generalized and "meaningful" expected value; however, consider the following issue. Suppose for n∈ ℕ, set A&sube;ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; B* is the set of all Borel measurable function in ℝᴬ for all A&sube;ℝⁿ, and B** is the set of all f∈B* with a finite-valued expected value—w.r.t the Hausdorff measure—then B** is a shy subset of B*. Hence a "positive measure" of Borel measurable functions needs to have a finite expected value to increase the chance that everywhere, surjective f:A→B has a finite expectation. To fix this issues, we wish to find a unique and "natural" extension of the expected value—w.r.t the Hausdorff measure—on bounded functions to unbounded/bounded f, which takes finite values only, so B** is a non-shy subset of B*. Note, we haven't found evidence suggesting mathematicians have thought of this problem; however, it's assumed, in general, there's no meaningful way of averaging functions which cover an infinite expanse of space. Note, we haven’t found evidence suggesting mathematicians thought of this problem; however, it’s assumed, in general, there’s no meaningful way of averaging functions which cover an infinite expanse of space. Regardless, we’ll choose a sequence of bounded functions using a "choice function". Note, we find the "choice function" using a question with criteria in §2.4. Also, in §3 and §4, we attempt to answer this question that should "choose" a sequence of bounded functions that a) meaningfully averages everywhere, surjective functions and b) obtains a finite average from a "positive measure" of Borel measurable functions.

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Section: Introductionmentioning

confidence: 99%

“…the main question of §3. 4) where the answer to the focus [3] is E * * [ f , F ⋆⋆⋆ k ] in eq. 48-using def.…”

mentioning

confidence: 99%