# 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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### Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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### Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the height. ...
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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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### If $P$ and $Q$ are Lebesgue partitions then is $P\cap Q$ a refinement to $P$ and $Q$?

My teacher said that if $P$ and $Q$ are Lebesgue partitions then $P\cap Q$ is a refinement to $P$ and $Q$. But shouldn't $$P\cup Q$$ be the refinement to both $P$ and $Q$? Or, are both $P\cup Q$ and ...
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### Is right this application of Hadamard three-lines theorem for $\frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
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### Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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### An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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### How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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### Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
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### Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
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### Expectation of a random vector and Fubini's theorem

I have a question related to the definition of expectation of a random vector, in particular, to its relation (if any) with Fubini's Theorem. Consider the random vector $X:=(X_1,X_2,X_3)$ of ...
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### Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
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### Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
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### Integration over a variety

If $M$ is a differentiable manifold equipped with an Atlas $\mathcal{A} = ( U_i , \varphi_i )_{ i \in I}$, we can then calculate the integral of a differential form $\omega$ over $M$ with the ...
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### Upper Lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable. Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...
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### Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
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### Proving translational invariance of Lebesgue integral

I am asked to show that the Lebesgue integral is invariant under translations. Specifically, Let $(\mathbb{R}, \Sigma, \mu)$ be a measure space, and for any $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
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### Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
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### Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done:  \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = \left(\sum_{k=1}^\...
Let $\{f_n\}$ be a sequence of functions in $L^2(0,1)$ such that $\lim_n f_n = f$ pointwise and $\vert f_n(x) \vert \leq g(x)$ for some integrable function $g$. By the dominated convergence theorem it ...
Problem: Let $X_{1},\cdots,X_{n}$ be samples from $X$. If $E\left|X\right|^{a}<\infty$ for some $a>0,$ and $n,k,r$ satisfies $r\leq a\cdot\min\left(k,n-k+1\right),$ then \$E\left|X_{\left(k\...