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Register a new userReal-valued measurable cardinals and sequentially continuous homomorphisms
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Authors
Vladimir Uspenskij
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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.
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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 $mathcal{M}(X,mathcal{A})$.
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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With a complete Heyting algebra $L$ as the truth value table, we prove that the collections of open filters of stratified $L$-valued topological spaces form a monad. By means of $L$-Scott topology and the specialization $L$-order, we get that the algebras of open filter monad are precisely $L$-continuous lattices.
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