Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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A Lebesgue integral with a $0$ denominator

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
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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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On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
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Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
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Bound on integrable nonnegative function $F$ given inequality with compactly supported continuous functions.

Full Question: Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that $\int_\mathbb R Ff \leq C\int_\mathbb R f$ whenever $f$ is a ...
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Different approaches to differentiability in $L^2$

We can use different approaches to differentiability of $L^2(\mathbb{R})$ functions, e.g. we can say that $f\in L^2(\mathbb{R})$ is differentiable iff $f$ has a differentiable version (representative)....
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Where $\{q_n\}=\mathbb Q$ and $f_n:[q_n-2^{-n-1},q_n+2^{-n-1}]\to[0,\infty)$ with $\int f_n\,d\lambda=1$, show $\sum_{n=1}^\infty f_n<\infty$ a.e.

That is: Let $\mathbb Q=\{q_n\}_{n\in\mathbb N}$ be an enumeration of the rationals. Let $f_n$ be a nonnegative Borel measurable function supported on $q_n\pm 2^{-n-1}$ with $\int f_n\,d\lambda =1$, ...
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