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Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

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1answer
19 views

Convergence to zero in $L^{p}(\mathbb{R}^{N})$

Is true that if $u_{n}$ converges to $0$ in $L^{p}(\mathbb{R}^{N})$ for some $p\geq 1$ then $u_{n}\rightarrow 0$ in $L^{q}(\mathbb{R}^{N})$ for any $q\geq 1$? My guess is that the above statement is ...
0
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1answer
26 views

Integral with respect to measures

Suppose there are two non-negative random variables X and Y with their cumulative distribution functions $F_X$ and $F_Y$. Further, we know $F_X(t) \leq F_Y(t)$ for all $t>0$. Then, for a non-...
2
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1answer
50 views

Lebesgue integral of ${x^2}$

I am having trouble calculating the Lebesgue integral of ${x^2}$ over ${]0,1]}$. I followed the instructions from my college lessons and constructed a sequence ${f_n}:=\sum_{k=1}^{n*2^n}\frac{k-1}{2^n}...
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1answer
20 views

Applying the dominated convergence theorem back and forth

Suppose $(X,\mathcal{M},\mu)$ is a measure space, $(f_{n})$ and $(g_{n})$ two sequences of integrable functions that tend to $f(x)$ and $g(x)$, respectively. Suppose also that $$|f_{n}g_{n}|\leq h_{1}...
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0answers
10 views

Reference for “continuity in mean” of integrable functions

In the measury theory class I'm taking we proved the following theorem Let $f \in L^1(\mathbb{R^n})$. Then for almost every $x \in \mathbb{R^n}$ it holds that $$ \lim_{r\downarrow 0} \frac{1}{V_r}\...
1
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1answer
25 views

Proving equality of Lebesgue integral if $f(\Omega) \subseteq \mathbb{N}$

Let ($\Omega,F,\mu$) be a finite measure space and $f: \Omega \rightarrow [0,\infty]$ be a nonnegative measurable function. Prove that : If$f(\Omega)\subseteq \mathbb{N}$ then $ \int fd\mu= \sum_{n=1}...
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0answers
21 views

Proof verification on integral function and absolute continuity

I've tried to prove something I was interested in, hope someone could spend a couple of minutes to review it. I'm sorry if it is too long, but I tried to be as clear as possible. Let $f:[0,+\infty)\...
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0answers
19 views

$G(x)=\int_\mathbb{R}e^{-|x-y|}f(y)dy$ is in $L^2(\mathbb{R})$ [on hold]

Let $f\in L^2(\mathbb{R})$, and why is $\int_\mathbb{R}e^{-|x-y|}f(y)dy$ is in $L^2(\mathbb{R})$ too ?
1
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1answer
34 views

How can I show $\int_{U}\frac{\partial f}{\partial x_{i}}(x)g(x)d\lambda^{d}(x)=-\int_{U}f(x)\frac{\partial g}{\partial x_{i}}(x)d\lambda^{d}(x)$

Let $U \subseteq \mathbb R^{d}$ and $f \in C_{c}^{1}(U)$ while $g \in C^{1}(U)$ Show that: $\int_{U}\frac{\partial f}{\partial x_{i}}(x)g(x)d\lambda^{d}(x)=-\int_{U}f(x)\frac{\partial g}{\partial ...
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0answers
31 views

Fubini Thm to prove $\int _{[0,\infty)^n \setminus [0,1]^n} \frac{1}{\sum_1^n x_i^{a_i}} d\lambda <\infty \Leftrightarrow \sum_1^n\frac{1}{a_i} <1$

This question have an answer which doesn't use Fibini-Tonelli here: A problem in n-dimesnsional Lebesgue measure The proposition states that $a_1,a_2,...,a_n >0$: $\int _{[0,\infty)^n \setminus [...
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1answer
31 views

Bartle The Elements of Integration exercise 6S

Let $f_n$ $\in$ $L_p(X,\chi,\mu)$, $1 \leq p <+\infty$ and let $\beta_n$ be defined for $E \in \chi$ by $$\beta_n(E) = \left(\int_{E} |f_n|^p d\mu\right)^{1/p}$$ and suposse that $(f_n)$ is a ...
1
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1answer
51 views

Non-zero continuous function $f(x)$ such that $\int_0^1x^k (1-x)^{n-k} f(x)dx=0$ for every $k=0,1,…,n$ and $n$ is a non-negative integer.

Does there exists a non-zero continuous function $f$ such that $\displaystyle \int_0^1 x^k (1-x)^{n-k} f(x)dx=0$ for every $k=0,1,2,...,n$ where $n$ is a non-negative integer. EDIT: The problem is ...
0
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1answer
20 views

Is the lebesgue integral of a measurable function continuous?

I was wondering if the lebesgue integral of a measurable function at least continuous? What kind of regularity on the integrand do we need for it to be absolutely continuous so that we can say its ...
1
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1answer
24 views

Integral of continuous function over a triangle

Let $D\subset\mathbb{R^2}$ a triangle which has the corners $(0,0),(1,0),(0,1)$ and $g: \mathbb{R} -> \mathbb{R}$ continuous. Then $\int_Dg(x+y)dL^2(x,y)=\int_0^1tg(t)dt$ where $L^2$ is ...
1
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0answers
21 views

Approximation of an $L^1-$ function of two variables by trigonometric polynomials.

We know as a theorem that the trigonometric polynomials are dense in $L^1([0,1))$ For instance for a Lebesgue integrable function we use the Fejer kernel $$F(x)=\sum_{n=-N}^N(1-\frac{|n|}{N+1})e^{2\...
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0answers
23 views

Why does Riemann Integration equal Lebesgue Integration in Probability theory [on hold]

In one of our recent tutorials on Probability Density Functions, I was confused that we said if $f: \mathbb R \to \mathbb R$ is Riemann Integrable $\Rightarrow$ Then $f$ is Lebesgue Integrable. $(1)$ ...
1
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1answer
33 views

Is the squared difference between a $L^{2}$-function and a Non-$L^{2}$-function in $L^{2}$?

Let $(\Omega, \mathcal{A}, P)$ be a probability space. Furthermore, let $Y,V : \Omega \rightarrow \mathbb{R}^n$ be random vectors and let $$ \int_{\Omega}{\vert \vert Y \vert \vert ^2}dP < \infty$...
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0answers
19 views

Laplace functional for a point process of exceedances

Let $$\mathit N_n(\cdot)=\sum_{i=1}^n \varepsilon_{\frac in}(\cdot)\mathit I_{(\mathit X_i>u_n)}$$ be a point process of exceedances where $(\mathit X_n)$ is a sequence of random variables and $(...
2
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2answers
25 views

Bounded sequence in $\mathcal{L}^p$ implies $\mathcal{L}^p$ convergence if sequence converges a.e

I'm working on $\mathcal{L}^p(E,\mathcal{A},\mu)$ space and I would like to prove the following: Let $p\in [1,\infty[$ and $\mu(E)< \infty$, suppose also that i) $f_n \to f, \mu$ a.e ii) there ...
2
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1answer
29 views

Continuous distribution-valued function induces distribution

Suppose that the map $\mathbb{R}^n \to \mathcal{D}'(\mathbb{R}^n), \hspace{3mm}\eta\mapsto E_\eta$ is continuous. Furthermore let $\mu$ be a Radon-measure with compact support. I'm having trouble ...
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0answers
33 views

Norm on Variable Lebesgue space

Please how to start the prove of (7.27), I need hint or the idea to begin please
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0answers
13 views

How do I determine which $V$ to exclude to obtain a Diffeomorphism

In order to use the Lebesgue Transformation Formula, I need to find a $V$ to exclude, but I am unsure of how exactly to do it. Example: Spherical Coordinates $\Phi: \mathbb R_{>0 }\times]0,\pi[\...
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1answer
117 views

Prove Lebesgue integrability for an (almost) trigonometric function

Let $X = [0,1]$, $\mathfrak{M}$ - is a $\sigma$ algebra of Lebesgue measurable subsets of $X$, $\mu$ - Lebesgue measure on $\mathfrak{M}$ Function $f:X\to\mathbb{R} $ is defined as: $f(x) = \sin{nx}$ ...
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0answers
57 views

Prove Lebesgue integrability and calculate

Let $X = [0,1]$, $\mathfrak{M}$ - is a $\sigma$ algebra of Lebesgue measurable subsets of $X$, $\mu$ - Lebesgue measure on $\mathfrak{M}$ Function $f:X\to\mathbb{R} $ is defined as: $ f(\frac{1}{n}) = ...
2
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0answers
27 views

Help in understanding the proof of a claim made in the proof of the Beppi Levi Theorem in Schilling.

In the proof of the Beppo Levi theorem in Schilling the following claim is made: $$f\leq u\Rightarrow I_\mu(f)\leq\sup_{j\in\mathbb{N}}\int u_j d\mu.$$ The proof of this claim is not hard but there ...
3
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2answers
42 views

Exchange limit on bounds of Lebesgue integral

Let $(E_n)_{n \in \mathbb{N}}$ be a sequence of measurable sets such that $\lim_{n \to \infty} E_n =E$ for some measurable set $E$. When does it hold that $$ \lim_{n \to \infty}\int_{E_n}f d \mu = \...
2
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1answer
30 views

Show that this is an inner product

Let's define $$(f,g)=\int_{\mathbb{R}} \frac{f(x)\bar{g}(x)}{1+x^2}dx$$ $\forall f,g\in X=\{h:\mathbb{R}\rightarrow\mathbb{C}:$ $h$ is Lebesgue-measurable and bounded over $\mathbb{R}$} I have to ...
3
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2answers
25 views

Beppo Levi's theorem, is this assertion correct?

My notes report the following assertion for the theorem: Beppo Levi's Theorem: Let $E$ be a measurable set and $\{ f_n(x)\}$ a sequence of integrable functions on E, such that $\lim\limits_{n\to\...
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0answers
18 views

Convergence in the topology of $L^2_\text{loc}$ implies convergence in $B^2$?

Let $f_n$ be a sequence of functions in $L^2_\text{loc}(\mathbb{R})$ which converge to a function $f\in L^2_\text{loc}(\mathbb{R})$ in the topology of $L^2_\text{loc}(\mathbb{R})$, i.e., $f_n\to f$ in ...
0
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1answer
40 views

A case where Lebesgue integrable implies Riemann integrable

Let $I$ an interval on $\mathbb{R}$ such as $I=(a,b)$, with $a$ or $b$ could be equal to infinity. And we have $f\in \mathcal{L}^1(I,\mathcal{B}(I), \lambda)$, then do we have always $$\int_{(a,b)}...
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1answer
24 views

How to check that a random variable is in $L^p(\Omega,F,\mathbb{P})$

How to check that a random variable is in $L^p(\Omega,F,\mathbb{P})$? Maybe I should provide an example to get my point across. Let $X_\alpha$ be a random variable with CFD $F_\alpha(x) = (1 - \...
3
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2answers
302 views

When is a Lebesgue integrable function a Riemann integrable function?

When is a Lebesgue integrable function a Riemann integrable function ? And if we have $f\in \mathcal{L}^1([0,1],\lambda)$, does it implies that $f$ is Riemann integrable, and why ?
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0answers
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Problem on Dominated converges theorem [closed]

$f_n(x)$= $n$ if $1/n^3≤x≤8/n^3$ And $0$ if $0≤x≤1/n^3$ or $8/n^3<x≤1$ If $|f_n(x)|≤M(x)$ where $M(x)$ is integrable then prove that $M(x)$= $2/x^⅓$ if $0<x≤1$ and And $0$ if $x=0$
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1answer
37 views

A $\sigma$-finite measure on $[0,1]$

Let $f:[0.1] \to \Bbb{R}$ such that $$f(x)=\sum_{n=1}^\infty \frac{1}{n^2}\frac{1}{\sqrt{|x-r_n|}}$$ where $\{r_1,r_2...r_n...\}=\Bbb{Q} \cap [0,1]$ We define the measure $$μ(Ε)=\int_E f^2 dx$$ for ...
3
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0answers
13 views

Question about integral notation in a Markov process + how to evaluate said integral

I'm reading Chapter 11 of Puterman's book on Markov Decision Processes (in particular, about continuous-time Markov processes). There's a lot of notation involved, but I've tried to distill the ...
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0answers
36 views

Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

I must show that Let $f$ be Lebesgueable integrable, then $$\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)|dy = 0$$ for almost every $x \in \mathbb{R}$. Tentative proof: $$|f(x) - f(y)| \geq ...
0
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0answers
23 views

integration with respect to probability measure [closed]

I would like to calculate (a part of) the area of a simplex in n-dimensional space. I know that the integration with respect to the probability measure would be the result. However, I do not know how ...
1
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2answers
22 views

Example for $f \in \mathcal{L}^q(\mathbb{R}, \lambda) \setminus \mathcal{L}^p(\mathbb{R}, \lambda)$ for $q < p$

We consider $q, p \in [1, \infty)$. I'm required to show that $\mathcal{L}^p(\mathbb{R}, \lambda) \subsetneq \mathcal{L}^q(\mathbb{R}, \lambda)$ and $\mathcal{L}^q(\mathbb{R}, \lambda) \subsetneq \...
1
vote
1answer
40 views

Varying definition of monotone convergence

My current lecture states the theorem of monotone convergence: Let $(X, \mathcal{E}, \mu)$ a measure space and $\varphi_n: X \rightarrow \mathbb{R}$ an increasing sequence of $\mu$-integrable ...
0
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3answers
116 views

How to prove this limit is $1/4$

$$\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{\int_{0}^{1}{\cdots \int_{0}^{1}{{{\left( \frac{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}{n} \right)}^{2}}d{{x}_{1}}d{{x}_{2}}\cdots d{{x}_{n}}}}}$$ I ...
0
votes
1answer
16 views

Show $\int_{\mathbb R^{d}}f(\alpha x)d\lambda^{d}(x)=\frac{1}{|\alpha|^{d}}\int_{\mathbb R^{d}}f(x)d\lambda^{d}(x)$

Let $\alpha \neq 0$ Use $|\alpha|^{d}\lambda^{d}(B)=\lambda^{d}(\alpha B)$ $(*)$ to: Show that $\int_{\mathbb R^{d}}f(\alpha x)d\lambda^{d}(x)=\frac{1}{|\alpha|^{d}}\int_{\mathbb R^{d}}f(x)d\lambda^...
1
vote
0answers
19 views

Hints on $\lambda^{d}(\alpha B)=|\alpha|^d\lambda^{d}(B)$

Let $B \in \mathcal{B}^{d}$ and $\alpha\neq 0$ Any ideas on how to show: $\lambda^{d}(\alpha B)=|\alpha|^d\lambda^{d}(B)$ My idea using "simple sets": Let $B:=[a_{1},b_{1}[\times...\times[a_{d},b_{...
0
votes
2answers
19 views

Uniform convergence of $f_n$ implies convergence in $\mathcal{L}^1([0,1])$

Let $f_n \in \mathcal{L}^1([0,1])$ for all $n\in \mathbb{N}$ then it follows that $f_n \rightarrow 0$ in $\mathcal{L}^1([0,1])$. The proof goes as follows: Uniform convergence means i): $||f_n-0||_\...
0
votes
1answer
24 views

Determine $\lambda^{2}(\{(x,y) \in \mathbb R^{2}: x \in \mathbb Q\cap [0,1], y \in [0,1]\})$

Define $M:=\{(x,y) \in \mathbb R^{2}: x \in \mathbb Q\cap [0,1], y \in [0,1]\}$ Find $\lambda^{2}(M)$ Ideas: Well, first I should show that $M \in \mathcal{B}^2$ Note that for $x \in \mathbb Q \...
3
votes
2answers
198 views

Lebesgue Integral, an example

I have been trying to studying the construction of Lebesgue integral for a while now. I am following the Princeton Lectures on Analysis and I am stuck at the part where it defines the integral of non-...
2
votes
0answers
23 views

Monotone Convergence Theorem in calculating Lebesgue Integral

Given the Lebesgue measure $m_2$ on $(\mathbb{R}^2,\mathbb{B}_2)$, I have the following integral: $$\mu(\mathbb{R}^2)=...=\int 1_{[1,\infty )}\frac{1}{x^2}dm(x)$$ My professor insist on using the ...
1
vote
2answers
57 views

Expectation of $XY$ bounded for all bounded $Y$ implies $X$ is $L^p$

I'm trying to prove: Let $X$ be a real random variable, $p, q \in (1,\infty)$, $\frac 1 p + \frac 1 q = 1$. If there is $C < \infty$ such that $|\mathbb E[XY]| \leq C ||Y||_q$ for any bounded ...
0
votes
1answer
44 views

Lebesgue integrable function involving distance function

$f$ is a bounded measurable function with compact support in $\mathbb{R}, \int_\mathbb{R}f(x)dx=0$ $M_f(x)=\sup_{t>0}\{|\int_\mathbb{R}f(y)\frac{t}{t^2+(x-y)^2}dy|\}$ Prove $M_f(x)\in L(\mathbb{R}...
-1
votes
0answers
25 views

Prove the equality (measure theory and Lebesgue-integrability)

Let $(X,\mathbb{M},\mu)$ be a measurable space with a complete countably additive measure and let $f:X\to \mathbb{R}$ be Lebesgue-integrable. $A_n \in \mathbb{M}, A_{n+1}\subseteq A_{n}, n\in \mathbb{...
1
vote
1answer
28 views

An application of Lebesgue Differentiation Theorem

Let $\delta_0$ be the measure defined as $\delta_0(A)=1$ if $0 \in A$ and $\delta_0(A)=0$. Let $g_h(x)=\frac{1}{h}1_{[-h,h]}$ and define $\nu_h$ as: $\nu_h(A)=\int_{A}g_h(x)dx$. Show that for every ...