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Register a new userThe Real-Valued Bochner integral and the Modern Real-Valued Measurable function on $mathbb{R}$
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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.
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A.V.Arkhangelskii asked in 1981 if the variety $mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $Gto H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequential if and only if its local weight is not a $g$-sequential cardinal. The product of a family of non-trivial $g$-sequential topological groups is $g$-sequential if and only if the cardinal of this family is not $g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is $g$-sequential if and only if its weight is not a $g$-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.
Two separated realcompact measurable spaces $(X,mathcal{A})$ and $(Y,mathcal{B})$ are shown to be isomorphic if and only if the rings $mathcal{M}(X,mathcal{A})$ and $mathcal{M}(Y,mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $mathcal{M}(X,mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $mu$ on $(X,mathcal{A})$, the $m_mu$-topology on $mathcal{M}(X,mathcal{A})$ is 1st countable if and only if it is connected and this happens when and only when $mathcal{M}(X,mathcal{A})$ becomes identical to the subring $L^infty(mu)$ of all $mu$-essentially bounded measurable functions on $(X,mathcal{A})$. Additionally, we investigate the ideal structures in subrings of $mathcal{M}(X,mathcal{A})$ that consist of functions vanishing at all but finitely many points and functions vanishing at infinity respectively. In particular, we show that the former subring equals the intersection of all free ideals in $mathcal{M}(X,mathcal{A})$ when $(X,mathcal{A})$ is separated and $mathcal{A}$ is infinite. Assuming $(X,mathcal{A})$ is locally finite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of $mathcal{M}(X,mathcal{A})$.
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.
In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space
In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions ${psi_{lambda}}_{lambdain Lambda}subset L^2(mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-valued function $f in L^2(mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${{|f ast psi_lambda|}_{lambda in Lambda}}$. We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $infty$, it suffices to know $|f ast psi_lambda|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $L^2(mathbb{R}^d)$, $d=1,2$, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.
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