# Questions tagged [measure-theory]

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

1,606 questions

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### 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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338 views

+50

### 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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119 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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94 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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104 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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**1**answer

99 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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33 views

### 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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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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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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131 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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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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64 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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182 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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**1**answer

99 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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44 views

### Identical push-forward but not stationary

I'm having some trouble coming up with a counter-example for this problem:
Give an example of a stochastic process $\{X_n : n \in \mathbb{Z}^+\}$ on $(\Omega, \mathcal{F}, P)$ such that $P_{X_n} = P_{...