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Register a new userMeasurable Cones and Stable, Measurable Functions
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Added by
Michele Pagani
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Thomas Ehrhard
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We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.
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A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory.
The set of all maximal ideals of the ring $mathcal{M}(X,mathcal{A})$ of real valued measurable functions on a measurable space $(X,mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $mathcal{M}(X,mathcal{A})$ in terms of the points of $hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $mathcal{M}(X,mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = mathcal{M}(X,mathcal{A})$ where $mathcal{A}$ is the $sigma$-algebra consisting of the zero-sets of $X$.
Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-on
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 $mathcal{M}(X,mathcal{A})$.
A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that approximately continuous functions are Baire 1., i.e., pointwise For any f:R^2 -> R define f_x(y) = f^y(x) = f(x,y). A function f:R^2 -> R is separately continuous if f_x and f^y are continuous for every x,y in R. Lebesgue in his first paper proved that any separately continuous function is Baire 1. Sierpinski showed that there exists a nonmeasurable f:R^2 -> R which is separately Baire 1. In this paper we prove: Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous and f^y is Baire 1 for every x,y in R. Then f is Baire 2. Thm 2. Suppose there exists a real-valued measurable cardinal. Then for any function f:R^2 -> R and countable ordinal i, if f_x is approximately continuous and f^y is Baire i for every x,y in R, then f is Baire i+1 as a function of two variables. Thm 3. (i) Suppose that R can be covered by omega_1 closed null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 2 for every x,y in R. (ii) Suppose that R can be covered by omega_1 null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 3 for every x,y in R. Thm 4. In the random real model for any function f:R^2 -> R if f_x is approximately continuous and f^y is measurable for every x,y in R, then f is measurable as a function of two variables.
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