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.

learn more… | top users | synonyms

7
votes
0answers
296 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
330 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
125 views

Theorem $2.14$ page $40, 41$ in Rudin - Real and Complex Analysis

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
6
votes
0answers
71 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
80 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
votes
0answers
75 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
votes
0answers
451 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
votes
0answers
1k 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
votes
0answers
127 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
votes
0answers
70 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
votes
0answers
201 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
votes
0answers
129 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
votes
0answers
51 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
votes
0answers
28 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
votes
0answers
37 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
40 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
votes
0answers
49 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
votes
0answers
81 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
votes
0answers
87 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 ...
3
votes
0answers
84 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
156 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
votes
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
80 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
233 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
373 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
8 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
9 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
21 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
votes
0answers
26 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
votes
0answers
37 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
44 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
46 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
59 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
53 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
60 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 ...
2
votes
0answers
45 views

Show that $d+1$-dimensional Lebesgue measure of set $G$ equals $0$

Let $D \subset \mathbb{R}^d$ and let $f:D \rightarrow \mathbb{R} $ be measurable function. Let $G=\{(x_1,x_2,\ldots,x_d,f(x_1,x_2,\ldots,x_d))\in \mathbb{R}^{d+1}:(x_1,x_2,\ldots,x_d)\in D \} $ be the ...
2
votes
0answers
48 views

Riemann and Lebesgue improper integral Proof

I've been trying to find some notes on the following statement: Let $f:(a,b] \to \mathbb{R}$, $f\geq 0$, and $f\in\mathcal{R}[a+\epsilon , b]$ for any $\epsilon>0$. Then $\int_a^bf=\lim_{\epsilon ...
2
votes
0answers
30 views

zeros of the dyadic maximal function

Recall the definition of the Hardy-Littlewood maximal function $Mf$ (https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function). If we replace the balls in the definition by dyadic cubes ...
2
votes
0answers
48 views

Lebesgue integral over $\mathbb R^2$ of the function $f(x,y)=2(x-y)e^{-(x-y)^2}\chi_{\{x>0\}}$

Let $f:\mathbb R^2\to \mathbb R$ be given by $$f(x,y)=\begin{cases}2(x-y)e^{-(x-y)^2}& \text{ if }x>0 \\0&\text{ otherwise}\end{cases}$$ Given that $\int^\infty_{-\infty} e^{-z^2} dz=\sqrt ...
2
votes
0answers
81 views

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 $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
2
votes
0answers
33 views

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 ...
2
votes
0answers
41 views

Lebesgue integrability of $x^a \sin(x)$ on $[0,\infty[$

Why is $x^a \sin(x)$ Lebesgue-integrable on $[0,\infty[$? It's obviously measurable, so why $\int |f|<\infty$... You can divide the integral by the period of sine, but I don't know how to cope with ...
2
votes
0answers
21 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 ...
2
votes
0answers
43 views

Are these functions Lebesgue integrable?

let's consider the function $$f: [0,1] \to \mathbb{R}^+, \quad f(x) = \begin{cases} x^{-a} & x \in \mathbb{Q} \; \text{and} \; x>0\\ 0 & \text{otherwise}. \end{cases}$$ for some $a \geq ...
2
votes
0answers
25 views

$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$ is differentiable?

Let $ D=\{(x,y) \in \mathbb{R}^2 : x>0, y>0\}$. Show that the function $$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$$ is well defined on $D$. ...
2
votes
0answers
71 views

Lebesgue-Stieltjes integral w.r.t. measure defined by absoluting continuous $F$

I know that if $F:[a,b]\to\mathbb{R}$ is a non-decreasing absolutely continuous function then$$\int_a^b f(x)dF(x)=\int_a^b f(x)F'(x)d\mu$$where the first integral is the Lebesgue-Stieltjes integral ...