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        Questions tagged [measure-theory]

        Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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        65 views

        Duality mapping of a the space of continuous functions? [on hold]

        The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...
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        320 views
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        A question concerning Lusin’s Theorem

        We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in $...
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        votes
        1answer
        118 views

        Volume form under holomorphic automorphisms

        $(M,\omega)$ is a compact Kaehler manifold and $f_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V_s$. My question is whether the function $\frac{f_{t,s}^* \omega^n}{\omega^n}$ ...
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        votes
        1answer
        93 views

        If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?

        Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of ...
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        votes
        1answer
        77 views

        Equivalent notion of approximate differentiability

        Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one? $$\lim_{r \to 0} \rlap{-}\!\!\int_{...
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        1answer
        94 views

        Finitely additive measure on Cartesian square of countable set

        Let $\mu$ be a probability measure on $(\omega, 2^\omega, \mu)$ measure space which is finitely additive and $\mu(A)=0$ for finite sets. We can define as usual $\mu^2$ on semiring $\mathcal{G}=\{A\...
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        Formal justification of the Chaos game in the Sierpinsky triangle

        I want to justify why the Chaos game works to produce Sierpinsky triangle. I use a theorem taken from Massopoust Interpolation and Approximation with Splines and Fractals. Suppose that $(X,d)$ is a ...
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        2answers
        121 views

        Box dimension of the graph of an increasing function

        This Hausdorff dimension of the graph of an increasing function shows that: Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
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        0answers
        59 views

        Functional characterization of local correlation matrices?

        Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
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        50 views

        Relation between the set measures of real number and the real coordinate space of finite dimensions [closed]

        Consider a compact interval $[a,b]\subset \mathbf{R}$. Let $\Delta_1=\Delta([a,b])$ be the set of all Borel probability measures over $[a,b]$. Consider a Natural number $N\in \mathbf{N}$. What is ...
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        129 views

        Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions

        Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
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        1answer
        186 views

        Compactness of set of indicator functions

        Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
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        votes
        0answers
        63 views

        Is there a T3½ category analogue of the density topology?

        Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology?([1]) but for category (and meager sets) instead of ...
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        1answer
        181 views

        Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?

        Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have $\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...
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        vote
        1answer
        97 views

        Measurable function

        Let $S$ be a countable set. Consider $X=S^{\mathbb{N}\cup\{0\}}$ the topological Markov shift equipped with the topology generated by the collection of cylinders. Denoted $\mathcal{B}$ as the Borel $\...

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