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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7
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343 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
7
votes
0answers
343 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
6
votes
0answers
73 views

Various integration theories

Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely: Riemann Integration Riemann-Stieltjes Integration ...
5
votes
0answers
50 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
5
votes
0answers
100 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...
5
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0answers
85 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
5
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0answers
483 views

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 ...
5
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2k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
4
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129 views

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 ...
4
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0answers
74 views

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 ...
4
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103 views

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 ...
4
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204 views

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 ...
3
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0answers
57 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...
3
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62 views

Solution of $\int_0^{\pi} \frac{ y \cos y}{s^2+y^2} dy$

Is there a solution for the following integral (even in terms of Bessel or Struve functions)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
3
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133 views

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 $1\leq p< \infty$ ? ...
3
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57 views

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 ...
3
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31 views

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 $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
3
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0answers
41 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
3
votes
0answers
42 views

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 ...
3
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0answers
50 views

Convergence in Measure, Different Definitions

Let $(X, \mu)$ be a measure space, $E \subseteq X$ measurable, and $f_n$ a sequence of measurable functions on $E$. If $f$ is another function on $E$, I have seen two definitions for what it means ...
3
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0answers
25 views

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 ...
3
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0answers
95 views

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 ...
3
votes
0answers
56 views

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 ...
3
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0answers
87 views

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 ...
3
votes
0answers
164 views

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 ...
3
votes
0answers
37 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
3
votes
0answers
81 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
3
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0answers
87 views

Show: $\int f\, d\mu=\sum\limits_{y\in f(X)}y\cdot\mu(f^{-1}(y)))$

Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function which only takes a countable number of values. Show that $$ \int f\, d\mu=\sum\limits_{y\in ...
3
votes
0answers
82 views

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 ...
3
votes
0answers
235 views

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 ...
3
votes
0answers
381 views

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 ...
2
votes
0answers
32 views

Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$ where the $f_n$ are measurable functions? ...
2
votes
0answers
57 views

Calculate $\lim\limits_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx$

The problem is to find an integrable function that bounds $f_n(x) = \frac{n^p x^r \log x}{1 + n^2 x^2}$ where $r>0$, $p<\min \{2,1+r\}$ so we can calculate $$\lim_{n \to \infty} \int_0^1 ...
2
votes
0answers
42 views

minimal distance betwen a point and and the halfspace containing a convex set

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and $$\min_{f\in S} \|f-g^*\|$$ has the unique solution $f^*\in S$. ...
2
votes
0answers
56 views

If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
2
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0answers
31 views

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

$A$ is a invertible $n \times n$ matrix. Show: $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$ Can someone give me a hint on how to show that?
2
votes
0answers
53 views

Convergence of a subsequence of a subsequence of distribution functions

I'm trying to find a solution for the following problem: Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of signed (Baire)-measures (of bounded variation) on $[a,b]$ and let $F_{\mu_n}(t):=\mu_n([a,t))$ ...
2
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0answers
11 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
2
votes
0answers
13 views

$L_2((-2,2))$ function that has $L_1((-1,1))$ discrete derivative but not derivative

I am trying to find an example of a function $u\in L_2((-2,2))$ such that $||\delta_h(u)||_{L_1((-1,1))}$ is uniformly bounded in $0<|h|<1/2$ but $u'$ is not in $L_1((-1,1))$. Where ...
2
votes
0answers
38 views

Fourier transform and $L^1,$ $L^2$ convergence

Let $\phi \in L^2(\mathbb{R})$ and $\hat{\phi}$ be the Fourier transform of $\phi.$ Does this mean that $\sum_{m \in \mathbb{Z}} |\hat{\phi}(x + 2 \pi m)|^2$ converges in the $L^1$ sense on each ...
2
votes
0answers
22 views

Sion Minmax theorem for integral operators

Suppose $f, g\in S=L^p([0,1],\Sigma,\mu,[0,1])$. The objective $L:S\times S\to R$ is given by $$L(f,g)= \int f (h-g) d\mu, $$ where $h\in S$ is fixed. Could we apply Sion Minmax theorem to conclude ...
2
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0answers
28 views

Enigma in applying Lebesgue dominated convergenege theorem

Let $p\in \mathbb{C} \lbrack z],~p=p\left( re^{it}\right) ,n=\deg p$ and we want to compute de limit $$ \lim_{r\rightarrow\infty}\frac{1}{2\pi}\int_{0}^{2\pi}\frac{p^{\prime}\left( re^{it}\right) ...
2
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0answers
38 views

What does it mean when $\sigma$ is discriminatory?

I am reviewing this paper on the approximating power of neural networks and I came across a definition that I could not quite understand. The definition reads: where $I_n$ is the $n$-dimensional ...
2
votes
0answers
103 views

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 ...
2
votes
0answers
46 views

If a simple function is measurable, then is its characteristic function

I am working on a problem looks like this: If a simple function $s$ is measurable, show that its characteristic function $\mathcal X_{X_i}$ is measurable. Here are the ways I have been working ...
2
votes
0answers
59 views

Question about definition of the Lebesgue integral of a non-negative function

I am reading Royden's Real Analysis to learn about Lebesgue integration. Royden first shows that a bounded function on a set of finite measure is Lebesgue integrable if and only if it is measurable. ...
2
votes
0answers
67 views

Convergence of functions in $L^1$ implies convergence of derivatives $ a.e. $?

Update: Someone gives me a good counterexample, which basically answers all the questions I posed. The example is, $$f_n(x)=\frac{sin(nx)}{n}$$ ----------------------------------------- I just came ...
2
votes
0answers
63 views

Use of Fubini's Theorem

Let $f(m,n)$ be a real valued function for all $m,n\in\mathbb{N}$. Suppose that $$ \sum_{m=1}^\infty\lvert f(m,n)\rvert\le\frac{1}{n^2} $$ for each positive integer $n$. Use Fubini's Theorem to prove ...
2
votes
0answers
75 views

Computation of a Lebesgue-Stieltjes integral

I am asked to compute the integral $\int_{(0,3a]}x\,dF(x)$ with $a > 0$ where $$F(x) = \begin{cases} \pi & 0\leq x < a\\ 4+a-x & a\leq x < 2a \\ (x-2a)^2 & ...
2
votes
0answers
73 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...