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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23 views

Indicator and simple functions verify Fubini's Theorem (Lebesgue-Integral)

I'm working on a proof of Fubini's Theorem. The theorem says: Given $A\times B\in \mathcal{L}\times\mathcal{L}$ a Lebesgue measurable set in $\mathbb{R^2}$, and $f:A\times > B\rightarrow\...
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1answer
49 views

Show the sequence $f_n(x)=\frac{1}{n}\chi_{[0,n]}$ has no weakly convergent subsequence in $L^1$.

Show the sequence $f_n(x)=\frac{1}{n}\chi_{[0,n]}$ has no weakly convergent subsequence in $L^1[R]$. My observations: Assume $f_{n_k} \to f$ weakly, then: The sets where $f$ is positive or where it is ...
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1answer
30 views

Does the convergence to 0 in $L^2(0,T;L^2(K))$ for all compact $K \subset \mathbb{R}^{d}$ imply the convergence in $L^2(0,T;L^2(\mathbb{R}^{d}))$?

Let $(f_n)$ be a sequence in $L^2(0,T;L^2(\mathbb{R}^{d}))$ such that: $\|f_n\|_{L^2(0,T;L^2(\mathbb{R}^{d}))} \leq C_T$ for all $n \in \mathbb{N}$; $f_n \rightarrow 0$ a.e. in $(0,T)\times \mathbb{R}...
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1answer
64 views

Lebesgue integral of $f$ is equal to the Lebesgue measure of the area under the curve of $f$

I am self-studying measure theory, and I am trying to prove the following statement: Definition 1.3.2 (Simple function) We call a function $f:\mathbb{R}^d\to\mathbb{C}$ simple iff $f = c_1 1_{E_1} + \...
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1answer
31 views

Convergence to 0 of certain integral by DCT

I need to prove the following property: Let $f:\mathbb{R}^N\to \mathbb{R}$ a integrable function in $B(0,1)$. Then it is satisfied that $$\lim_{\varepsilon\to 0}\int_{|x|<\varepsilon}f(x)dx=0.$$ ...
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2answers
34 views

How is the product of two Lebesgue integrable functions?

Let $(X, \cal{A}, \mu)$ be a measure space and $f, g : X → \mathbb{R}$. Determine if the following implications hold in general: (i) both functions $f$ and $g$ are integrable $⇒ f \cdot g$ is ...
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1answer
2k views

Show a function is Lebesgue integrable

Hi I am struggling with a question but really I am struggling more with the concepts behind it so any help would be appreciated. Q/ Let $$f(x)=\begin{cases} x^{-\frac{1}{2}} & ,x\in(0,1) \\ 0 &...
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1answer
29 views

Applications of dominated convergence theorem for Lebesgue integrals

I have been working through measure theory, specifically the dominated convergence of Lebesgue integrals and its applications such as differentiating under the integral sign. There I came across the ...
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1answer
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How is the sum of two Lebesgue integrable functions?

I'm practicing for the final exam in real-analysis and I am at the chapter Measure Theory and Integration. I found this exercise, but I don't know how to solve it...Could you please help me? Let $(X, \...
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48 views

Prove that lim $\int_{E_n}f d\mu = 0$

Well they give me the following statement: Be $f$ integrable in $(X,F,\mu)$ and {$E_n$} $\subset F$ as {$E_n$} $ \downarrow E$ with $\mu(E) = 0$. Prove that $\lim_{n \to \infty} \int_{E_n} f d\mu = 0$...
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1answer
52 views

Explicit solution of wave equation in 3D using spherical coordinates

I'm trying to find the explicit solution of the wave equation in three spacial dimensions with the initial condition: $$ u(x,0) = \begin{cases} \sqrt{1-x^2}, & \text{if } |x| \leq 1, \\ 0, & \...
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2answers
27 views

Determine Lebesgue integral of a function containing floor function

I'm practicing for the real-analysis exam and I've got stuck at this integral... Could you help me, please? Determine: $$ \int_{[0,\infty)}\dfrac{1}{\lfloor{x+1}\rfloor\cdot\lfloor{x+2}\rfloor}d\...
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1answer
346 views

A Non-Counterexample to the Fubini Theorem with Counting and Lebesgue Measures

Let $X = Y = [0,1].$ Let $\mathcal{B}$ denote the Borel $\sigma$-algebra. Let $m$ denote the Lebesgue measure on $[0,1]$, and let $\mu$ denote the counting measure on $[0,1].$ Prove that $D = \{(x,y) :...
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Lebesgue integration on product domain from integrability of components and symmetry

Let me change the formulation somewhat: Suppose that $\Omega$ is a bounded open domain in $\mathbb{R}^n$, and that $f:\Omega\times\Omega\to\mathbb{R}$ is a Lebesgue measurable function. We know that: (...
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19 views

non-existence of homeomorphism between open subsets of $\mathbb{S}^{n}$ and $B(0,1,\mathbb{R}^{n})$

In the proof of no retraction theorem for $C^{2}$ function I have the necessity to justify why can't exist a homeomorphism between open subsets of $\mathbb{S}^{n}$ and $B(0,1,\mathbb{R}^{n})$. This ...
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1answer
40 views

Continuity of Lebesgue Measure of Continuous Correspondences

Let $X$ be a compact, convex subset of $\mathbb R^n$ with a non-empty interior and endowed with the sup-norm and let $C:X \rightrightarrows X$ be a continuous (i.e. upper and lower hemicontinuous), ...
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39 views

Find a strict bound of finitely many integrals $\int\frac{|h_jf-\lambda(h_jf)|^2}q\:{\rm d}\lambda$

Let $(E,\mathcal E,\lambda)$ be a measure space $f:E\to[0,\infty)$ be $\mathcal E$-measurable with $\lambda f<\infty$ (as usual $$\lambda g:=\int g\:{\rm d}\lambda$$ for $g\in\mathcal L^1(\lambda)$...
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0answers
20 views

Show $\int_{\mathbb{R}} |f(x)|dx = \int_0^\infty m(\{x:|f(x)| > t\})dt$ [duplicate]

Let $f\in L^1(\mathbb{R})$, and let $m$ denote lebesgue integration. Show $\int_{\mathbb{R}} |f(x)|dx = \int_0^\infty m(\{x:|f(x)| > t\})dt$. I'm honestly not sure how to get a good hold on this ...
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1answer
74 views

Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$

Theorem $:$ Let $f : [a,b] \longrightarrow \Bbb R$ be a Riemann integrable function. Then $f \in L_1[a,b]$ and $$\int f\ d\lambda = \int_{a}^{b} f(x)\ dx.$$ The proof given in my book is as follows $:...
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3answers
50 views

Let $f$ be a continuous function on $\mathbb{R}$ satisfying $\int_\mathbb{R}|f(x)|dx<\infty$. Can we conclude that $\sum_\mathbb{Z}|f(k)|<\infty$?

Let $f$ be a continuous function on $\mathbb{R}$ satisfying $$\int_\mathbb{R}|f(x)|dx<\infty.$$ Can we conclude that $$\sum_\mathbb{Z}|f(k)|<\infty?$$ Note: Continuity is necessary otherwise $f=\...
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1answer
18 views

If $X_n$ converges to $X$ in $L_p$ and $Y_n$ converges to $Y$ in $L_p$ then $X_n + Y_n $ converges to $X + Y$ in $L_p$ [closed]

I want to show that if $X_n \xrightarrow{L^p} X$ and $Y_n \xrightarrow{L^p} Y$ then $X_n + Y_n \xrightarrow{L^p} X + Y$ ($p \geq 1)$. My idea is to use the following facts (whose proofs I won't give ...
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20 views

Find the integral of $\int\limits_0^{2\pi } {Q\left( {f(\theta )} \right)Q\left( {g(\theta )} \right)d\theta } $?

I am trying to find the integral of the following function: $\int\limits_0^{2\pi } {Q\left( { - (e*\cos (a + \theta )*sqrt(x) + g} \right)*Q\left( { - (f*\sin (a + \theta )*sqrt(x) + h} \right)d\theta ...
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1answer
47 views

Twice continuously differentiable function $f:\Bbb R^2\to \Bbb R$ such that $f, f_{xx}, f_{yy}\in L^2$ is bounded

Suppose $f:\Bbb R^2\to \Bbb R$ is twice continuously differentiable, and that $f, f_{xx}, f_{yy}\in L^2$. Then is it true that $f$ is bounded? I first tried to find a counterexample, but I couldn't. ...
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2answers
89 views

Is $f(x)=\sum_{n=1}^{\infty}\frac{x}{1+n^2x^2}$, $x\in[0,1]$ continuous on $[0,1]$

Let $f(x)=\sum_{n=1}^{\infty}\frac{x}{1+n^2x^2}$, $x\in[0,1]$. Question: Show that $f$ is Lebesgue integrable and determine whether $f$ is continuous on $[0,1]$. For the first part, I have no problem, ...
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0answers
42 views

The law induced by a norm $\rho$ on $\mathbb{R}^n$ with respect to Lebesgue's measure.

For any function $\phi:\mathbb{R}^n\rightarrow\mathbb{R}$ and set $A\subset\mathbb{R}$, let $\{\phi\in A\}$ denote $\{x\in\mathbb{R}^n:\phi(x)\in A\}$. Statement of the problem: Suppose $\rho$ is a ...
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1answer
18 views

inverse of the modulus in $L^p$

We all know that, given any bounded open set $\Omega\subset \mathbb R$ $$\frac{1}{x}\in L^p(\Omega)\iff p<1 \qquad \frac{1}{x}\in L^p(\Omega^c)\iff p>1$$ what do these conditions become for a ...
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3answers
125 views

Showing $\lim_{n\to\infty}\int_{\mathbb{R}} f(x)f(x+n) dx=0$

Problem Let $f(x)\in L^2(-\infty,\infty)$. Prove that $$ \lim_{n\to\infty}\int_{-\infty}^\infty f(x)f(x+n) dx=0. $$ My attempt Let $N\in\mathbb{N}$ and $f_N(x):=f(x)\chi_{[-N,N]}(x)$. For any $n\in\...
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1answer
85 views

Proof of the change of variables formula without using the Monotone Convergence Theorem

I recently encountered the problem Exercise 36 in Tao's An Introduction to Measure Theory. The link of an online version of this problem is here. Now I quote this problem as follows: Exercise 36 (...
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1answer
36 views

Dominated convergence theorem to find $\lim_{n\rightarrow\infty}\int_1^\infty\frac{n^2x^2}{1+n^4x^4}dx$ [duplicate]

Use dominated convergence theorem to find $$\lim_{n\rightarrow\infty}\int_1^\infty\frac{n^2x^2}{1+n^4x^4}dx$$ I know that the dominating function is $1/x^2$ now how can we apply Dominated convergence ...
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28 views

Generalization of the concept of derivatives and the fundamental theorem of calculus

Studying Lebesgue's integrals I wonder whether there exists such a thing as a generalisation of the fundamental theorem of calculus, with a generalisation of the concept of derivative, which would ...
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1answer
42 views

A basic property of Lebesgue integrability

Problem: $f$ is integrable on $\mathbb{R}$, prove there exists $h_n\to0$ where $h_n\in(0,1)$ such that $f(x+h_n)\to f(x)\ a.e\ x\in\mathbb{R}$ I tried to prove $\sum_{n=1}^\infty\mid f(x+h_n)-f(x)\mid&...
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0answers
33 views

Lebesgue integral on $[0,1]$ and alternating intervals

I am working on the following problem from some old exam: For $\ n>0, \ $ a set $\ A_n \subset [0,1] \ $ is defined as $$A_n \ = \ \bigcup_{k=1}^{2^{n-1}}\Bigg[\frac{2k-2}{2^n},\frac{2k-1}{2^n}\...
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1answer
55 views

Proof of countable additive property of Lebesgue Integrable functions

While self studying proof of Theorem 10.49 (Ch- Lebesgue Integral) , I am unable to prove it by myself as proof is not given in textbook (left as exercise). My attempt-> (a) it is clear that if f ...
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2answers
44 views

Show that $Mf_n(x) \nearrow Mf(x)$, where $f_n(x) \nearrow f(x)$ almost everywhere

I am trying to show that $Mf_n(x) \nearrow Mf(x)$, where $f_n(x) \nearrow f(x)$ almost everywhere $x$, $f_n(x)$ and $f(x)$ are nonnegative and locally integrable, and $$ Mf(x) = \sup_{r > 0}\frac{1}...
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1answer
31 views

Direct proof that integral of a function does not depend on the $\sigma$-algebra used to define it?

If $\mathcal{G}\subset\mathcal{F}$ are two $\sigma$ algebras on a set $X$, $\mu$ is a nonnegative measure on $(X,\mathcal{F})$ and $f:X\to[0,+\infty]$ is $\mathcal{G}$-measurable, then there are two ...
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2answers
81 views

To show $\lim_{x\rightarrow 0} x\ln x$ by using DCT?

I know $\lim_{x\rightarrow 0} x\log x=0$ can be proved by using l'Hospital, but I heard that this statement can also be shown by using dominated convergence theorem. Hint says we use the function $f(t,...
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1answer
44 views

Using General Lebesgue Dominated Convergence Theorem [duplicate]

Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ and $f$ is integrable over $E$. Show that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\...
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0answers
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Change of variable theorem for non invertible linear transformation

I'm working out the exercises in Folland's Real Analysis, and problem 2.54 asks how much of the theorem (below) would hold true if the linear map T is non invertible. I think we would need to use T on ...
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1answer
31 views

How to check $f(x) = x^{-\alpha}, x \in (0,1]$ is Lebesgue integrable?

Let $f(x) = x^{-\alpha}, ~ x \in (0,1], ~ \alpha \in \mathbb{R}$, then how can we show that $f$ is Lebesgue integrable? I can show that $f$ is measurable but I don't know how to proceed further. I ...
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1answer
26 views

Convergence of the sum constructed by approximation of the integral of an $L^1$ function

Let $f \in L^1( \mathbb{R}^n)$. Does the sum $$S(x) = \sum_{k \in \mathbb{Z}^n} f(k+x) $$ converge for almost every $x$? Intuitively I'm approximating the integral (which is finite), so I think this ...
2
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1answer
35 views

Integration of gauss curvature on tubular surface

Consider a tubular surface $S$, i.e. a surface which has a parametrisation $$F:I\times\mathbb{R}\rightarrow \mathbb{R}^3,\quad F(t,\varphi)=c(t) + r · \Big(\cos \varphi · n(t) + \sin \varphi · b(t)\...
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2answers
23 views

Relation between Riemann integrable and Lebesgue integrable functions

Let $f$ be Riemann integrable function and $g$ be Lebesgue integrable function on $[0,1]$. If $\int_{0}^{1} |f-g| = 0$, then what we can say about $g$ i.e. whether $g$ will be Riemann integrable or ...
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1answer
19 views

Question in Proof of theorem 11.22 Apostol mathematical analysis

While self studying mathematical analysis from Tom apostol I have 2 question I proof of above mentioned theorem. Question 1: How does author deduces $g_{x} $ is measurable on $\mathbb{R}$ ? Question ...
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0answers
26 views

nonnegative measurable function (monoton convergence theorem)

Let {$a_n$} be a sequence of non-negative sequence of real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. Show that $\int_Ef=\sum{a_n}$. The summation ...
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1answer
37 views

Lebesgue points

Lebesgue points are defined by: $$\lim_{r \to 0}\frac{1}{Leb(B(x,r))}\int_{B(x,r)}|f(y)-f(x)|dy=0 \text{ for a.e. } x \in \mathbb{R} \text{ w.r.t. } Leb$$ where $f\in L^1(\mathbb{R}), B(x,r)$ is the ...
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1answer
31 views

Show that $\lim_n f_n(x) = f(x)$ almost surely for an integrable function

An exercise taken from Klenke: Let $f \in L^1(\lambda)$, where $\lambda$ is the restriction of the Lebesgue measure to $[0,1]$. Let $I_{n,k} = [k2^{-n},(k+1)2^{-n}), $ for $n \in \mathbb{N}$ and $k = ...
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0answers
16 views

Notation used in integration in stochastic processes

In studying stochastic processes, I have come across the Lebesgue integral applied to a probability space, i.e. $$\int_M f d\textbf{P}$$ But I have also come across the notation $$\int_M f \textbf{P}(...
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1answer
53 views

Lebesgue integrable for function

$f: \mathbb{R} \to \mathbb{R}$, $$f (x) = \left\{\begin{array}{ll} \frac{x^2}{(e^x-1)^2} & x> 0\\ 0 & x \leq 0\end{array}\right.$$ Can the function be integrated Lebesgue in the measure ...
3
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1answer
46 views

Making intuition rigorous that integral of some positive function on set should be monotone in the Haar measure of the set

Let $\mathcal{M}$ be a compact Riemannian manifold with geodesic distance function $d$ and $\Omega$ its volume measure. Pick some $A,B\subseteq\mathcal{M}$ such that $\Omega(A)\ll\Omega(B)$, but: (1) $...
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1answer
21 views

Proper use of DCT with double integral [closed]

Let $f_n(x,y)\to f(x,y)$ pointwise where $f_n$ is a sequence of real-valued functions. If $|f_n(x,y)|\leq c$ where $c>0$ is constant, then can I use the dominated convergence theorem to justify $$ \...

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