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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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### Singular distribution F such that convolution F and F is an absolutely continuous distribution?

F is a singular distribution function concentrated on the positive half-line. Is it possible that 2-fold convolution F*F is an absolutely continuous distribution function? Please, give me an example.
1 vote
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### A different way to try to define a measure on the unit-circumference circle

Let C denote the Lie group (ℝ/ℤ, +), and let ℤn denote the subgroup of C generated by [1/n]. Let (X,n) be an ordered pair where X ⊂ C is an arbitrary subset, and n ∊ ℤ+, such that that the set of ...
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### Injectivity of two sided Laplace transform

Let $\mu,\nu$ be finite Borel measures on $\mathbb R$. Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:  \int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
• 379
1 vote
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### Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras"). In another post, it was explained ...
30 views

### Name for a regularity property of $\sigma$-ideals

Let $X$ be a topological space and let $\mathcal{B}$ be its Borel $\sigma$-algebra. Suppose $\mathcal{N} \subset \mathcal{P}(X)$ is a $\sigma$-ideal, i.e. $\emptyset \in \mathcal{N}$ and it is closed ...
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### Question on density of certain set of matrices

Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...
• 151
1 vote
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### Regularity of $\sigma$-finite measure pushforwarded by completion

Let $(X, d)$ be a metric space. Let $\mu$ be a $\sigma$-finite measure defined on borel subsets of $X$. Let $i \colon X \to \hat{X}$ be an isometry on image, where $\hat{X}$ is a complete metric space ...
314 views

### Integrating on orbits of algebraic groups

Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...
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### Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141. For simplicity I restae the ...
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1 vote