# 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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### 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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### $q$-integral of product of two functions

How to solve $\int\limits_a^b \left[\int\limits_t^b (x-tq)_{\Re(\alpha n+\beta-1)} d_qx\right]|\phi(t)|d_qt$ and what will be the answer? Where $|q|<1$ and the function $\phi$ be in the space ...
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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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### 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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### Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
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### Measurability of $f:X\times Y\to\mathbb{K}$ and $f(-,y):X\to\mathbb{K}$

Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of ...
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### Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
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### Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
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### Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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### Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
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### Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
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### Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$\int_{D(0,R)}|f(w)|dm(w)$$ and ...
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### $1 = \int f(x) \ dx$, by definition, or by Lebesgue's theorem?

We have that (in the context of Lebesgue integration)$$\lim_{n \rightarrow \infty} \int_{-\infty}^n f(x) \ dx = 1$$ I wish to show that this implies $\int_{-\infty}^\infty f(x) \ dx = 1$. Is this true ...
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### Showing Lebesgue's definition of measure implies measurability

19. Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of $E$ to be $\mu_*(E) = \mu_0(X) - \mu^*(E^c)$. Then $E$ is ...
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### Integration by parts for Dirac measure

We know that Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if x \in A}\\ 0 &\text{if x \notin A}\\ \end{cases}$$ We know that $\int_a^b f(y) \, d\delta_x(y)=f(x),$ if ...
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### How to use dominated convergence on $\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x)$?

I find it hard to find an appropriate dominating function for the integral $$I:=\lim_{t \to \infty}\int_{(0, \infty)} \frac{1-e^{-tx}(x \ sin t + \cos t)}{1+x^2}d \lambda_1(x), \ t > 0$$ ...
### Mathematical expectation of $F_\xi(\xi)$
Consider $F$ as a distribution function of some random variable $\xi$. The problem I'm trying to solve is to find integral: $$\int_{-\infty}^{+\infty}F(x)dF(x)$$ From what I see, there are two ways ...
### Averages of integral and$L^p$ space problem
Let $f: \mathbb R \to \mathbb R$ be an integrable function, for each $h>0$ let $$f_h(t)=\dfrac{1}{h}\int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ Suppose $f \in L^P$, prove the following (1) \$f_h ...