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Register a new userDefining the Mean of a Real-Valued Function on an Arbitrary Metric Space
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Authors
Kerry Michael Soileau
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We show how a metric space induces a linear functional (a mean) on real-valued functions with domains in that metric space. This immediately induces a relative measure on a collection of subsets of the underlying set.
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The like-Lebesgue integral of real-valued measurable functions (abbreviated as textit{RVM-MI})is the most complete and appropriate integration Theory. Integrals are also defined in abstract spaces since Pettis (1938). In particular, Bochner integrals received much interest with very recent researches. It is very commode to use the textit{RVM-MI} in constructing Bochner integral in Banach or in locally convex spaces. In this simple not, we prove that the Bochner integral and the textit{RVM-MI} with respect to a finite measure $m$ are the same on $mathbb{R}$. Applications of that equality may be useful in weak limits on Banach space.
In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics.
We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel-Lizorkin type.
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ varphi(n)$ be the Euler totient function. The result $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+Oleft ( x(log x)^{2/3}(loglog x)^{1/3}right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ sum_{nleq x}psi([x/n])=(15/pi^2)xlog x+O(xlog log x) $, and $ sum_{nleq x}sigma([x/n])=(pi^2/6)xlog x+O(x log log x) $ are also evaluated here.
We introduce and investigate the concept of harmonical $h$-convexity for interval-valued functions. Under this new concept, we prove some new Hermite-Hadamard type inequalities for the interval Riemann integral.
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