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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Show that $\lim_{x\rightarrow a}F(x)$ exists finitely for $\int_a^b|F'(x)|dx<\infty$, $F\in C^1$

Problem. Suppose $-\infty\le a<b\le\infty$ and $F:(a,b)\rightarrow R$ is a $C^1$ function such that$$\int_a^b|F'(x)|dx<\infty.$$ Show that $\lim_{x\rightarrow a}F(x)$ and $\lim_{x\rightarrow b}F(...
2 votes
1 answer
39 views

Show that $\int_{\mathbb R^d}fd\mu=\sum_{x\in C}f(x)$ for countable $C\subset R^d$

Problem. Let $C$ be a countable subset of $\mathbb R^d$ and $\mu$ be the counting measure on $\mathbb R^d$ i.e., $$\mu(A)=\#(A\cap C),\qquad A\in \mathcal B(\mathbb R^d).$$ Show that for a measurable $...
4 votes
1 answer
164 views

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 $$ ...
1 vote
1 answer
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Showing that $f_k = f \chi_{B(0,k)}$ has compact essential support.

Consider the usual Lebesgue measure on $\mathbb R^n$. Let $f$ denote a $p-$locally integrable function, that is, $$ \int_K |f(x)|^p \, dx < \infty, \quad \forall \text{ compact set } K \subset \...
0 votes
0 answers
35 views

How to show the following function is not Lebesgue integrable? [closed]

How to show that $\int_{||x||\ge 1} \frac{1}{||x||^n} dx $ where $x\in \mathbb{R}^n$ is not finite , i.e. $\frac{1}{||x||^n} $ is not integrable outside a ball of radius $1$ centered at $0$ ? I ...
4 votes
1 answer
5k views

Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuous function on $[a,b]$, let $\mu$ be a bounded Borel measure on $[a,b]$, and let $\Phi_\mu(t)=\mu([a,t))$ ...
1 vote
0 answers
50 views
+200

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 ...
3 votes
1 answer
2k views

Product of two Lebesgue measurable set is measurable

Let $A,B \subset \mathbb{R}$ two bounded and Lebesgue measurable sets. I have to show that $A\times B \subset \mathbb{R}^2$ is measurable and \begin{align*} \lambda(A \times B) = \lambda (A) \cdot \...
0 votes
0 answers
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Riemann integability of a bounded almost continuous function in $\mathbb{R}^d$, $d>1$

We know that a continuous function on a compact set in $\mathbb{R}$ is Riemann integrable if and only if the set of discontinuities of the function is at most countable. Suppose now we are in $\mathbb{...
1 vote
1 answer
32 views

Is there a need to specify $g$ as non-negative for $\int gd\nu=\int gfd\mu$ to hold?

Theorem. If $\nu$ has a density $f$ with respect to $\mu$, then $$\int gd\nu=\int gfd\mu$$ holds for non-negative $g$. Moreover, $g$ (not necessarily non-negative) is integrable with respect to $\nu$ ...
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Proof of $\int_{\Omega_2}f(y)\mu{\circ}T^{-1}(dy)=\int_{\Omega_1}f(T(x))\mu(dx)$

Theorem. Suppose $(\Omega_1,\mathcal A,\mu)$ is a measure space and $(\Omega_2,\mathcal A_2)$ is a measurable space. If $T:\Omega_1\rightarrow\Omega_2$ and $f:\Omega_2\rightarrow\bar{\mathbb {R}}$ are ...
3 votes
1 answer
73 views

Can we conclude that $f=g$ a.e. if $\int _Efd\mu =\int _Egd\mu $ for all measurable sets $E$?

Let $(X,\Sigma,\mu )$ be a measure space and $f,g:X\to\color{red}{[0,\infty ]}$ measurable functions. If $\int _Efd\mu =\int _Egd\mu $ for all $E\in \Sigma$, can we conclude that $f=g$ a.e.? I know ...
1 vote
1 answer
26 views

Existence of a sequence $\mu(I_j \cap I_k)= 0$ such that $\lim_{j\to \infty}\mu(I_j) = b-a$

Is it possible that given an interval $[a,b]$ with $a<b$, we can find a sequence of measurable subsets $I_j$ such that $\mu(I_j \cap I_k)= 0$ for $j \ne k$ while satisfying: $$ \lim_{j\to \infty}\...
1 vote
2 answers
82 views

Doubts in calculatating $\int_0^\infty \frac{n\sin x}{1+n^2x^2}\,\mathrm{d}x$

I have to calculate $$\int_0^\infty \frac{n\sin x}{1+n^2x^2}\,\mathrm{d}x$$ and I would like to use the DCT. I have said that $$\forall x\in (0,1)\qquad\frac{n\sin x}{1+n^2 x^2} \leq \frac{n\sin x}{2 ...
1 vote
1 answer
34 views

Lebesgue Integration, Epsilon Boundedness of the Integral of Function f Implying f is 0 Almost Everywhere

I was working on the following question from the University of Hawaii Real Analysis PHD Qualifying Practice Exam: $\text{Suppose } f \in L_1(\mathbb{R}) \text{ and there is } 0 < \varepsilon \text{...
2 votes
0 answers
52 views

How does the often used picture about Lebesgue integration relate to the actual definition

Often when comparing Riemann and Lebesgue integration pictures like the following are used (credits to this Reddit post by user u/Analemmath) However the actual most common definition of Lebesgue ...
0 votes
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Stein Analysis 3.5.2 - kernels with average zero

Consider the problem given below from Stein and Shakarchi's Real analysis. Here is the same question being asked before, but my attempt is different, so I wanted to ask as a new post. I want to know ...
0 votes
1 answer
26 views

$\int f(x) d\lambda^d(dx) = \int f(\phi(x)) |\mathrm{det}(\partial \phi)| d\lambda^d(dx)$ 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 +\...
4 votes
1 answer
2k views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
2 votes
1 answer
60 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 ...
20 votes
3 answers
9k views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
1 vote
1 answer
85 views

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 ...
0 votes
0 answers
32 views

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 ...
2 votes
1 answer
42 views

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\...
2 votes
0 answers
23 views

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| \ \...
1 vote
1 answer
38 views

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 ...
1 vote
0 answers
38 views

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. ...
0 votes
0 answers
39 views

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)=...
2 votes
1 answer
41 views

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 ...
3 votes
1 answer
1k views

Question regarding Lebesgue Integrability in $\sigma$ -finite spaces

I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable ...
4 votes
0 answers
92 views

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 $\mathbb{R}^n$ (...
0 votes
1 answer
27 views

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 ...
0 votes
1 answer
26 views

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}...
1 vote
1 answer
73 views

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\...
0 votes
1 answer
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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 ...
0 votes
0 answers
42 views

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}(...
6 votes
4 answers
400 views

Lebesgue integral question from wiki

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the Wikipedia entry on "Lebesgue integration" they define the Lebesgue integral as: Let ...
0 votes
0 answers
22 views

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?
2 votes
1 answer
41 views

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 \...
0 votes
0 answers
31 views

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 ...
2 votes
1 answer
61 views

$A$ 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 ...
1 vote
1 answer
103 views

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_{...
0 votes
1 answer
41 views

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. ...
4 votes
1 answer
1k views

The difference between Riemann integrable function and Lebesgue integrable function

My professor asked me how to intuitively understand Lebesgue's dominated convergence theorem and what's the effect of the integrable dominated function. More specifically, when we are given a Lebesgue ...
2 votes
1 answer
34 views

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 ...
2 votes
3 answers
1k views

Chebyshev's Inequality in proof of Proposition 23 of Royden?

This is from the 4th edition of Royden, on page 92. Proposition 23. Let $f$ be a measurable function on $E$. If $f$ is integrable over $E$, then for each $\epsilon >0$, there is a $\delta &...
0 votes
0 answers
23 views

$f_j \rightharpoonup f$ 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(...
1 vote
1 answer
88 views

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 $\mathbb{R}$. ...
2 votes
0 answers
24 views

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 \, ...
6 votes
1 answer
1k views

Counting measure on the power set sigma-algebra for the natural numbers

My textbook does not provide much about counting measures and integration. So I decided to set up integration on space $(N , P(N) , \mu_c ,R)$ myself, imitating the construction of the Lebesgue ...

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