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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Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything ...
xyz's user avatar
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Is $\sin x/x$ Lebesgue integrable over $[0,\infty)$?

Let $f(x)=\frac{\sin x}{x}$ if $x\neq 0$ and $f(x)=0$ if $x=0$. Is $f$ Lebesgue integrable? Also, is $\sin^2x/x^2$ Lebesgue integrable? Note that $\lim_{x\to 0}\frac{\sin x}{x}=1=f(0)$. So $f$ is ...
H.Y Duan's user avatar
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Use of dominated convergence theorem in Manski (1985)

I'm confused by the use of the dominated convergence theorem (DCT) in Lemma 5 of Manski (1985) (see below). Note that $b = (\tilde{b}_1, \dots, \tilde{b}_{K-1}, b_K).$ Specifically: I presume the ...
Giacomo's user avatar
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Analysis of an expression involving a function on $\mathbb R^n$. Related to limits, supremums and translations.

Let $n \in \mathbb N, \, 0 < \lambda < n$ and $1 \leqslant p < \infty$, consider the usual Lebesgue measure on $\mathbb R^n$ and define the function $f \colon \mathbb R^n \to \mathbb R$ by $$ ...
xyz's user avatar
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Interchange limit of an Lebesgue-Integral with the sequence of functions

I've got just a short question and I think an easy one but I am not able to answer it: If $(f_n)_n$ is a sequence of Lebesgue-integrable functions converging to a Lebesgue-integrable function $f∈L^1$ ...
RobRTex's user avatar
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How to use the continuity of the translation operator on $L^p(\mathbb R^n)$ in this specific question? (Lebesgue measure)

Consider the usual Lebesgue measure on $\mathbb R^n$. For context, in what follows I am using the definition $$ L^p_{\operatorname{loc}}(\mathbb R^n) := \{ f\colon \mathbb R^n \to \mathbb R \text{ ...
Temirbek Alikhadzhiyev's user avatar
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We know that $g \in L_1$ and that $g*g \leq g$ nearly everywhere. Show that: $\int_{\mathbb{R}} g(x) d\lambda_1(x) \leq 1$

$g \in L_1$ such that $g*g \leq g$ nearly everywhere. Show that: $$\int_{\mathbb{R}} g(x) d\lambda_1(x) \leq 1$$ From what is said, I know that: $$g*g \leq g \implies \int_{\mathbb{R}} (g*g) (x) \ d\...
thefool's user avatar
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Convergence a.e. plus convergence of $L^1$ norms implies convergence in $L^1$ [duplicate]

Let $(X, \mathcal{X}, \mu)$ be a measure space. Suppose $(f_n)_{n\in\mathbb{N}} \subset L^1(\mu)$ is such that: $f_n \to f \in L^1(\mu)$ a.e. $ \int_A | f_n | \ \mathrm{d} \mu \to \int_A |f| \ \...
Caio Lins's user avatar
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On the hypothesis of Fubini's theorem

I'm reading Bauer's Measure and Integration Theory. After the proof of Fubini's theorem in page 140 he introduces the following example: I don´t understand why the theorem doesn't apply in this case. ...
Student2271's user avatar
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Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
SiegAndy's user avatar
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Line integral with respect to the arc length

Find $$\int_{C}\phi\,ds$$ in the counterclockwise direction, where $\phi(x,y)=x+y$ and $s$ is the arc length. Here $C$ is the line joining $(1,0)$ and $(0,1)$. Solutions I tried: Sol 1: Let $\alpha(t)=...
Mathguide's user avatar
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Line integral of a vector field over a closed path

Let $f(x,y,z)=(y,x,x)$ be a vector field. Then it can be shown that $f$ is not a gradient. Find a closed path in $\mathbb{R}^3$ such that the integral of $f$ over that Parth is non-zero. Regarding ...
Mathguide's user avatar
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$\int f(x) d\lambda^d(dx) = \int f(\phi(x)) |\mathrm{det}(\partial \phi)| d\lambda^d(dx)$ for non-integrable $f$?

I have a minor issue with the transformation formula for integrals. It says if $\phi: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a diffeomorphism $f: \mathbb{R}^d \rightarrow \mathbb{R}\cup \lbrace +\...
Perelman's user avatar
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On the measurability of a certain function

Some techniqal question regarding the measurability of a function: Assume $f :\mathbb{R}^n \rightarrow \mathbb{R}$ is measurable. Clearly $(x,y)\mapsto x-y$ is measurable. My question is, is it ...
Perelman's user avatar
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Does this ersatz of Riemann-Lebesgue lemma hold?

Let us consider $f \in L^1(\mathbb{R^2})$. I'd like to know if one can apply some sort of Riemann-Lebesgue lemma to this quantity : $\forall k \in \mathbb{R}, \, I(k) = \int_{\mathbb{R}}f(x,k)e^{-ikx}...
Hugo's user avatar
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Construction of a simple function by open sets

A simple function $\phi: \mathbb{R} \to \mathbb{R}$ can be written as: $$\phi(x) = \sum_{k=1}^n a_k \chi_{E_k}(x)$$ where each $E_k$ is a measurable set and $\cup E_k = \mathbb{R}$ and $\chi_{E_k}$ is ...
MC2's user avatar
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Are two sets almost equal in this case?

Suppose I have $A$ and $B$ two unbounded, measurable, open sets in $\mathbb{R}^n$. I also have that $$\int_A x^{\alpha} - \int_B x^{\alpha} = 0$$ for every monomial $x^{\alpha}$ on $\mathbb{R}^n$ (...
Soumya Ganguly's user avatar
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Common Lebesgue integral vs upper and lower Lebesgue integral

Given a measure space $(X,{\cal A},\mu)$ there is the following definition of the Lebesgue integral: First one defines the integral for a simple function $g=\sum_{i=1}^n \lambda_i \chi_{A_i}$, $A_i\in ...
Pong's user avatar
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Unbounded function in $L^\infty$ implies the existence of a $L^2$ function for which the product does not belong to $L^2$

Let $a: \mathbb{R} \to \mathbb{C}$ be a measurable function which is not essentially bounded (So does not belong to $L^{\infty}(\mathbb{R})$). I need to show that there exists a function $f$ in $L^{2}(...
liamsi Meean's user avatar
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Use Fubini for $\int_0^\infty g(x)dx=\int_0^\infty \int_0^x f(x-y)g(y)dydx.$ [closed]

Let $f,g$ be integrable functions and $h(x)=\int_0^x f(x-y)g(y)dy$. Is there a way to calculate/simplify the expression $\int_0^\infty g(x)dx$? I started with $$\int_0^\infty g(x)dx=\int_0^\infty \...
Robert's user avatar
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integral of the $n$-th power of a function and its distribution

Let $f_1,f_2$ be two measurable functions on $(0,1)$. If thre exists an constant $c>0$ such that $$\int_0^1 f_1^nd m =\int_0^1 f_2^n dm =O((cn)^n) ,$$ then are $f_1$ and $f_2$ equimeasurable?
user92646's user avatar
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Find a $m>0$ such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right) $

Consider $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$ and $T_f$ is the regular distribution associated to $f$. Assume now $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$, nonnegative satisfying $T_f \in \...
YuerCauchy's user avatar
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If $\int_0^1 x^nf(x)=0$, then $f(x)=0$ a.e. [duplicate]

Suppose that $\int_0^1 x^nf(x)=0$ for all nonnegative integers $n$, where $f$ is a Lebesgue measurable function that is bounded. How do you prove that $f(x)=0$ a.e. on $[0,1]$. I've seen this problem ...
ruben's user avatar
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1 answer
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$A$ is a borel subset of $\mathbb{R}^2$ and we know that $\lambda_2(A) = \pi$. Prove that: $\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$

We assume that $A$ is a borel subset of $\mathbb{R}^2$ and that $\lambda_2(A) = \pi$. Prove that: $$\int_A (x^2 + y^2) \ d \lambda_2(x,y) \geq \frac{\pi}{2}$$ I see that it might be useful to use ...
thefool's user avatar
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Let $f(x) = x^{-1}\sin(x^{-1})+\cos(x^{-1})$. How can one prove that $\int f^+ = \infty$?

Hunter mentiones in his notes (p.44) that the function $f(x) := x^{-1}\sin(x^{-1})+\cos(x^{-1})$ lacks a defined Lebesgue integral, which must because both $f^+$ and $f^-$ have infinite integrals. ...
Sam's user avatar
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1 answer
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Prove that the function $f(x) = \frac{1}{x^p}$ belongs to $L_{1}(1,\infty)$ if and only if $p > 1$.

Prove that the function $f(x) = \frac{1}{x^p}$ belongs to $L_{1}(1,\infty)$ (Where $L_1$ is the space of functions that are Lebesgue integrable) if and only if $p > 1$. proof $\Rightarrow$ Suppose ...
a7777's user avatar
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$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Suppose that $\Omega \subset \mathbb{R}^d$ is bounded with a Lipschitz boundary and $f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$. Show that, for a subsequence, $\nabla(...
Mr. Proof's user avatar
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1 vote
1 answer
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Prove that $f = g$ almost everywhere on $\mathbb{R}$

Let $f$ and $g$ be functions in $L^1(\mathbb{R})$ such that $$ \int_E f \, dm = \int_E g \, dm $$ for every measurable subset $E$ of $\mathbb{R}$. Prove that $f = g$ almost everywhere on $\mathbb{R}$. ...
satl's user avatar
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prove that $\int_{A \cup B} f \, dm + \int_{A \cap B} f \, dm =\int_{B} f \, dm+ \int_{A } f \, dm$

Let $f$ be a non-negative measurable function. Prove that for $A$ and $B$ measurable subsets of $\mathbb{R}$, the following holds: $$ \int_{A \cup B} f \, dm + \int_{A \cap B} f \, dm =\int_{B} f \, ...
a7777's user avatar
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1 answer
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How to prove that $ {\displaystyle \sup_{x \in \mathbb R^n, \, r > 0} r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy < \infty}$?

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$. Consider the function $f_\alpha \colon \mathbb R^n \to \mathbb R$ defined by $$ f_\alpha(x) := \|x\|^{\frac{\lambda - n + \alpha}{p}} \chi_{...
xyz's user avatar
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1 answer
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About convergent sequence $f_n \to f$ in $L^p(U)$ ( Convergence in norm, passage of limit under integral etc.. ; Evans's PDE )

Let $U$ be a bounded, connected, open subset of $\mathbb{R}^n$. Assume $1 \le p \le \infty$ .Let $f_n \to f$ be a convergent sequence in $L^p(U)$. My question is, then, Q.1. $ \lim_{n\rightarrow\...
Plantation's user avatar
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How to do change of variable to polar coordinates in higher dimensions double integral?

Let $B$ be a ball of radius $R$ centered at $0$. I am considering the integral $$\int_B \int_B \frac{1}{|x-y|^k}dxdy,$$ where $k$ is some real number (which can make the integral infinity). To ...
Alvis Zhalovsky's user avatar
2 votes
1 answer
59 views

Lebesgue Integral Graphical Picture/Video

I'm teaching a real analysis course and I'm trying to find a video or picture or app that illustrates the Lebesgue integral of a nonnegative function as the "supremum of integrals of smaller ...
Joshua Isralowitz's user avatar
3 votes
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62 views

How to prove that ${\displaystyle \, \, \lim_{r \to 0} \, \sup_{x \in \mathbb R^n} \, r^{-\lambda} \int_{B(x,r)} |f_\alpha(y)|^p \, dy = 0}$?

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$. Consider the function $f_\alpha \colon \mathbb R^n \to \mathbb R$ defined by $$ f_\alpha(x) := \|x\|^{\frac{\lambda - n + \alpha}{p}} \chi_{...
xyz's user avatar
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Definition of Lebesgue integral in Stein

For non-negative functions, the definition of $\int f$ is consistent among Stein, Folland and Wikipedia. However, for real-valued functions, there's a difference. Stein (page 64) first defines ...
Daniel Mendoza's user avatar
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Integral estimate with Young's inequality

Consider two functions $g,h$ with $g\in L^m(\Omega)$ and $h\in L^{l+\epsilon}(\Omega)$, where $\Omega\subset\mathbb{R}^l$ ($l\geq 2$) is some bounded domain, $\epsilon>0$ is small (for simplicity ...
HelloEveryone's user avatar
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1 answer
37 views

Prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.

Following this question:$f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous. Our question is that let $f\in L_1(\mathbb{R})$. Prove $F(x) = \int_{-\infty}^x f(t) dt$ is ...
H.Y Duan's user avatar
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5 votes
1 answer
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Direct method with integral constraint

Let $\Omega\subset\mathbb{R}^n$ be nonempty, open and bounded with $C^1$ boundary. Let $p\in[1,n)$. Let $g\in C(\mathbb{R})$ satisfy $$|g(y)|\leq C(1+|y|^q)$$ for some $C<\infty$ and some $q$ with $...
hannah2002's user avatar
2 votes
2 answers
84 views

Show that $||f_n-f||_p \rightarrow 0$ as $n \rightarrow \infty$

Given $(X, \mathcal{A}, \mu)$ be a finite measure space and $f_n \in L^p(X, \mu)$ where $f_n(x) \rightarrow f(x)$ almost everywhere as $n \rightarrow \infty$ and $1 \leq p < \infty$. Now suppose $||...
The Limit Does Not Exist's user avatar
5 votes
1 answer
129 views

Proving the equality case of the Rising Sun Inequality

I'm stuck at the finish line on a pair of Exercises from Tao's Introduction to Measure Theory: Exercise 1.6.12 (Rising Sun Inequality). Let $f : \mathbb{R} \to \mathbb{R}$ be an absolutely integrable ...
Nick A.'s user avatar
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2 answers
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Lebesgue Integral help

I'm studying for my Real Analysis Masters Exam and the only thing that I don't understand at all is Lebesgue integrals and cant find any good examples. A old master exam problem I'm trying says: ...
Walter Wilde's user avatar
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With finite $\mu$, suppose $f_n$ converges in measure to $f$, and for all $n$, $||f_n||_2\leq 1$. Prove $||f_n-f||_1\rightarrow 0$

This is actually a part b), where part a) was to show $f\in L^2$. This was simple: as $f_n$ converges in measure to $f$, there exists a sub-sequence which converges $\mu$-a.e., thus Fatou's gives: $\...
cable's user avatar
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1 answer
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Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets?

Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$). ...
Michael's user avatar
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Schwarz rearrangement for a function with one variable fixed

Decompose Euclidian space as $\mathbb{R}^N = \mathbb{R}^{n} \times \mathbb{R}^{m}$. If $u \in H^1(\mathbb{R}^N)$ I know it is well defined the Schwarz rearrangement of $u$ and the Pólya–Szegő ...
Lucas Linhares's user avatar
1 vote
1 answer
25 views

Fixing a variable of a $H^1_0(\Omega)$ function

Let $\Omega \subset \mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. Suppose that $\Omega \subset A_1 \times A_2$, where $A_i \subset \mathbb{R}^{N_i}$. If $u \in C^\infty_0(\Omega)$, I ...
Lucas Linhares's user avatar
2 votes
0 answers
38 views

Measurability of an integral operator?!

Is it possible to prove the measurability of the following map $\Phi_n \colon (C(\mathbb{R}^d), \sigma(\mathcal{C})) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, $n \in \mathbb{N}$, defined by $$ \...
TrippyMushroom95's user avatar
2 votes
1 answer
31 views

A discussion on different versions of the Lebesgue Differentiation Theorem.

The simplest formulation of the Lebesgue Differentiation Theorem (LDT) (that I am aware of) is the following: LDT (Simplest formulation). Given a function $f \in L^1_{\text{loc}}(\mathbb R^n)$, we ...
xyz's user avatar
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Is there a framework where a derivative of a singular function makes sense?

Suppose we have a cumulative distribution function $F(x)$ Then by Lebesgue decomposition theorem, we may decompose $F$ into: absolutely continuous part, singular part, piecewise step part. As for the ...
Leon Kim's user avatar
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$fg \in L^1$ for every $f \in L^2$ implies $g \in L^2$

Suppose that $f g$ is in $L^1([a,b])$ for every $f$ in $L^2([a,b])$, I understand that this implies that $g$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it? This question was ...
Alessandro's user avatar
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0 answers
38 views

Everywhere existence of marginals

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a joint PDF, i.e., $\|f\|_{L^1}=1$, which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$. What is a necessary and sufficient condition under which the ...
Amir Sagiv's user avatar

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