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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1answer
26 views

Can this exercise be solved by DCT, I was only able to use MCT.

How would you solve this exercise? You don't need to give me the details, just the general idea. Let f be a Lebesgue integrable function. Show that $\int f(x+a) d\lambda=f(x) d\lambda$ and ...
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0answers
22 views

Measure of triangular area

Let $\lambda\in[0,1]$, $\Omega=[0,1]^2$, $\vec{m}$ and $\vec{n}$ be two linearly independent vectors, $i\in\mathbb{N}$ and $h(t)$ the periodic extension of $$\tilde{h}(t):=\begin{cases} (1-\lambda)t ...
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2answers
71 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
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2answers
56 views

What does $d\mathbb{P}(\omega)$ in integral mean?

What does $d\mathbb{P}(\omega)$ under integral sign mean? Like $$\int_B Xd\mathbb{\mathbb{P}}(\omega)$$ Can somebody explain? How can we integrate $X$ with respect to $\mathbb{P}(\omega)$ where ...
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0answers
10 views

Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
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2answers
237 views

It seems obvious, but how to prove it formally?

Let $f$ be a not decreasing, Lebesgue integrable function with finite integral over $[0,+\infty)$. It seems obvious to me that $\lim_{b\rightarrow\infty}\int_b^{\infty}f(x)\,dx=0$ then. But how can ...
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0answers
28 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
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1answer
30 views

Holder inequality (reverse or equality?)

For bounded $\Omega\in\mathbb{R}^n$, it is easy to see by the Holder inequality that $\int_{\Omega} u\,dx\leq (\int_{\Omega} 1^2\,dx)^{\frac{1}{2}} (\int_{\Omega} ...
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1answer
21 views

Lebesgue's integral and measure

I have got one question, because I see a big black hole in my knowledge about measure theory and Lebesgue's integrals: $$lim \int_A \sqrt [n] {x_1x_2} dl_2(x_1 x_2), A = {x_1^2 + x_2^2 <1, 0 \le ...
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1answer
37 views

lebesgue's integral and theorems [closed]

I have got an exercise from Lebesgue's integral: $$\lim_{n \to \infty} \int_{A} x^n y^{2n} \, dl_2 (x,y), \ A=\{ (x,y) \in \mathbb{R}^2 \mid 4x^2+y^2 \le 1 \}$$ I do not really understand Lebegues's ...
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0answers
31 views

Proof monotone convergence theorem, why do they use this lim sup?

I have a question about the proof of the MCT. First they use a lemma, this is ok, but I'll show it for completeness: Now comes the proof. But I am wondering, why do they use a lim sup here?, why ...
3
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1answer
68 views

Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Let $f(x)$ be a non-decreasing function on $[0, 1].$ You may assume that $f$ is differentiable almost everywhere. Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$. I am having a hard time with this ...
2
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1answer
38 views

Example with almost every convergence where the dominated convergence theorem fails

So I ran into this exercise, and I want someone to check the accuracy of my answer, because I feel pretty sure that I make some mistakes which I can't see. Let $f_n(x) : \mathbb{R} \to \mathbb{R}, ...
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1answer
24 views

(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...
3
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0answers
46 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 ...
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1answer
15 views

Proof of FTC, continuity part, for Lebesgue integrable functions

The part of the FTC I am interested in says: If $f$ is a Lebesgue-integrable function on $[a,b]$, then $F(x)=\int_a^xf(t)\,dt$ is continuous. This is usually considered a lemma or something for ...
6
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1answer
59 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
2
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1answer
36 views

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are both complete measure spaces. Consider the following two measure spaces: $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ and $(X ...
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2answers
40 views

Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Suppose $f$ is a real-valued (possibly infinite-valued) function on some measure space $(X, \Sigma, \mu)$, and suppose that it is measurable. Note that $f$ is not necessarily nonnegative. Does it ...
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1answer
27 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
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1answer
25 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
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1answer
28 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
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0answers
36 views

Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains an open interval [duplicate]

Suppose that $E$ and $F$ are Lebesgue measurable sets of $\mathbb{R}$, and their Lebesgue measures $m(E) > 0, m(F) > 0.$ Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains a nonempty ...
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1answer
16 views

Lebesuge integrable function always bounded?

Assume $f: [0,1] \to \mathbb{R}$ is Lebesgue integrable, does it imply that $f$ is bounded almost surly?
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3answers
92 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
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0answers
31 views

McShane vs. Henstock-Kurzweil: Lebesgue integrable

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
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1answer
20 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
5
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1answer
142 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
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1answer
37 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
2
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1answer
38 views

Is $\frac{\mathrm d}{\mathrm dx} \sin x/x = \cos x/x - \sin x/x^2$ Lebesgue integrable?

Is $$\frac{\mathrm d}{\mathrm dx} \frac{\sin x}{x} = \frac{\cos x}{x} - \frac{\sin x}{x^2}$$ Lebesgue integrable? In other words, is $$ \int_{\mathbb{R}} \left| \frac{\mathrm d}{\mathrm dx} ...
2
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1answer
44 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
3
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2answers
145 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
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1answer
67 views

If the integral $\int_0^\infty xf(x)\,dx$ converges, so does $\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx$

Let be $f:[0, \infty)\rightarrow [0,\infty)$ a measurable function such that $$\int_0^{\infty}x\cdot f(x)\,dx< \infty.$$ Show that $$\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx<\infty .$$ ...
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0answers
34 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
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0answers
46 views

Lebesgue-integrable and existence of integral

I have given the following function $$ f(x,y) = \begin{cases} 1 &, \ x \in \mathbb{Q} \\ 2y & , \text{ otherwise} \end{cases} $$ This is a measurable function in sense of Lebesgue. Now, I ...
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1answer
35 views

Integral $ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $ (Lebesgue)

I have to compute the following integral: $$ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $$ $\chi_{B_k}(x) =\begin{cases} 1 &, \text{if } x \in B_k \\ 0 ...
2
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2answers
56 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
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1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
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1answer
23 views

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$? What does it mean for $s$ to be integrable? 1. This is last minute exam revision. ...
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2answers
24 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
1
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1answer
56 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } ...
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1answer
29 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
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0answers
6 views

Images of intersections and Lebesgue integrals

How does the image not commuting with intersections, instead being just a subset of it $$f(U \cap V) \subset f(U) \cap f(V),$$ tell us that the Lebesgue integral is better than the Riemann integral? ...
0
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1answer
38 views

Absolute continuity and derivatives of integrals

I am preparing for a comprehensive at the end of the month, so I would appreciate any input I could get on this solution. I am pretty confident if the first part, but I think the second answer could ...
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0answers
43 views

Introduction of Lebesgue Theory

While reading Real Analysis- Stein, Shakarchi, I came up with following questions, some of which are in the "Introduction" of the book. The purpose of a series of questions is "to get actual ...
2
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2answers
102 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
0
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1answer
33 views

Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
0
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1answer
12 views

can representatives of an equivalence class of L_1(r^d)be nonmeasurable

Motivation: The composition of 2 Lebesgue measurable functions need not be measurable. This problem can be dealt with in a case by case basis( like with convolutions). Or as Big Rudin does, apply ...
3
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4answers
87 views

Can someone provide an example of Lebesgue integration on ordinary functions?

I'm trying to understand Lebesgue integral and integration but I'm completely mired by many author's motivation of this subject as only a tool for pathological functions such as the Dirichlet ...
2
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0answers
34 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...