# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
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### Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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### Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
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### Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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### Real Analysis, Folland The Dominated Convergence Theorem

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Question: 2.24 The Dominated Convergence Theorem - Let $\{f_n\}$ be a sequence in $L^1$ such that ...
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### Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)?

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)? This is from the last sentence in the proof in the following ...
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### Why is the bounded linear functional $T(g)=\int_X fg d\mu$ an isometry?

Royden claims the following in Real Analysis on page 400. T : $L^q$(X, μ) -> ($L^p$(X, μ))* is an isometry. Can anyone explain why is T a mapping from $L^q$(X, μ) to ($L^p$(X, μ))* instead of to ...
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### Is there a measure space $(X,\mathcal M, m)$ such that $\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$?

I have in mind the following question: Is there a measure space $(X,\mathcal M, m)$ such that the range of $m$ satisfies $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
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### Question on product measure: $\int_{[0,R]^2} g(x,y) df_1(x) \times df_2(y) = \int_{[0,R]} \left( \int_{[0,R]} g(x,y) df_1(x) \right) df_2(y)$ holds?

Suppose I have a real valued positive increasing functions $f_1(x), f_2(y)$. Then we know we can define Riemann-Stieltjes integral by defining measures $df_1(x)$ and $df_2(y)$. Let $g(x,y)$ be a ...
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### Real Analysis, Folland Corollary 2.19 Integration of Nonnegative functions

Corollary 2.19 - If $\{f_n\}\subset L^+$, $f\in L^+$, and $f_n\rightarrow f$ a.e., then $\int f \leq \liminf\int f_n$. Proof - We have that $\{f_n\}\subset L^+$, $f\in L^+$ and $f_n\rightarrow f$ a....
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Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\... 0answers 20 views ### Measurability of “conditional” pushforward measure Let$\Omega,\mathcal F, \mathbb P$be a standard probability space, and$T$,$R$compact spaces.$F:\Omega\times T \to R$a measurable function, which can be seen as a random function$T\to R$. Let$\...
What means "$\sigma$-algebra of all Borel sets" in this context? Is it Borel $\sigma$-algebra $\mathfrak{B}(X)$ or any $\sigma$-algebra $\mathfrak{M}$ containing $\mathfrak{B}(X)$?