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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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12 views

Lebesgue integration: integral of continuous function tends to infinity

I'm studying measure theory and Lebesgue integration and I've run into this problem: let $ f:R \rightarrow R$ be a continuous function such that: $\int f^+d\lambda_1 = \int f^-d\lambda_1 =+\infty $ ...
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1answer
19 views

Integral of average of integral equal to integral itself

I was trying to prove the following and had totally no idea how to start. Let $f$ be integrable over $(-\infty, \infty)$, and let $h> 0$ be fixed. Prove that $$\int_{-\infty}^{\infty} (\frac{1}{2h}...
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25 views

What is a tube in $\mathbb{R}^n$?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
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1answer
27 views

Prove that $\lim_{n\rightarrow \infty} \int_{[-n,n]} f\,d\lambda= \int f\,d\lambda.$

I am working on the following exercise: Let $\lambda$ denote Lebesgue measure on $\mathbf{R}$. Suppose $f:\mathbf{R}\rightarrow \mathbf{R}$ is a Borel measurable function such that $\int|f|<...
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1answer
12 views

Measure on a sigma-algebra with integral

Let $\mu$ be a measure on $(X, \mathcal{A})$ and $f:X \to \mathbb{R}, \ f \geq 0$. Define $\mu_f(E): \mathcal{A} \to \mathbb{R}, \ \mu_f(E):=\int_E f \ d\mu$ for $E \in \mathcal{A}$. How to prove ...
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1answer
19 views

Bogachev change of variables theorem.

I am reading the Bogachev's change of variable theorem proof. And I am stuck in an argument contained in it: Let $F:U\subset \mathbb{R}^n\to \mathbb{R}^n$ an injective $C¹$ map. Given $\epsilon>$ ...
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1answer
36 views

A “nice result” in Functional Analysis

Let $f:[0,+\infty)\to \mathbb{R}$ be a continuous function satisfying $f(x)\to 0$ for $x\to +\infty$. Prove that if $f\in L^1([0,+\infty))$ then $f\in L^p([0,+\infty))$ for all $p\ge1$. My attemp was:...
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1answer
30 views

Hypotheses on Plancherel's theorem

Plancherel's theorem is stated as (e.g. in Rudin's Real and Complex Analysis) If $f\in L^1 \cap L^2$ then $$ \|f\|_2 = \|\hat f\|_2 $$ where $\hat f$ is the Fourier transform of $f$. On the ...
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1answer
21 views

Calculating volume of a set $K=\{(x,y,z)^T \in \mathbb{R}^3:x^2+y^2+z^2 \le 4, x^2+y^2 \le 2x\}$

I want to to calculcate the volume, that is $\lambda_3$, of $K=\{(x,y,z)^T \in \mathbb{R}^3:x^2+y^2+z^2 \le 4, x^2+y^2 \le 2x\}$. I guess I need to evaluate a triple integral for this but I don't ...
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Lebesgue integral simple function query

I'm trying to show that $\int_{-\infty}^{\infty}\Big(\sum_{n=1}^{\infty}n\chi_{(\frac{1}{n+1},\frac{1}{n}]}\Big)d\mu = \infty$ where $\mu$ is the Lebesgue measure. Approach: I swap the sum and the ...
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1answer
18 views

Showing that the almost uniform limit of functions with bounded $L_\infty$ norms is in $L_\infty$

Suppose that $(f_n)$ is a sequence of functions for which there exists a finite constant $C$ such that the $L_\infty$ norm of $f_n$ is less than or equal to $C$ for all $n$. Suppose further that $f_n$...
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1answer
19 views

Hardy Littlewood maximal function $|h(x)|<|h^*(x)|$ for almost every x

Suppose $h: \Bbb R \to : \Bbb R$; Hardy Littlewood maximal function $h^*: \Bbb R \to [0, \infty]$ defined by $h^*(x)=\sup_{t>0}\frac 1{2t}\int_{x-t}^{x+t}|h|$. Now to prove $|h(x)|\leq|h^*(x)|$ ...
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Integral Transformation from circle to unit sphere

I want to show that $\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{|S^1|}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$ This is essentially a shift and dilation ...
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1answer
18 views

In a measurable partition of an interval, the sum of the measures of the subsets in the partition equals the length of the interval

I am working through A User-Friendly Introduction to Lebesgue Measure and Integration, by Gail S. Nelson. On page 67 Nelson defines a measurable partition of the interval $[a,b]$ to be finite ...
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1answer
24 views

$\int|f|^\alpha|g|^{1-\alpha}d\mu\le(\int|f|d\mu)^\alpha(\int|g|d\mu)^{1-\alpha}$

$f,g$ on $(\Omega, \mathcal{A}, \mu)$ and $0<\alpha<1$. Then $$ \int|f|^\alpha |g|^{1-\alpha} d\mu \le \left(\int|f|d\mu\right)^\alpha \left(\int|g|d\mu\right)^{1-\alpha} $$ This ...
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0answers
21 views

Non-product measure proof of Fubini’s theorem [on hold]

Can you state/direct me to a proof of Fubini’s theorem that does not rely on product measure ?
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11 views

Integral of inverse fuctions for noncontinuous distributions

Let $ F:[a,b]\rightarrow[0,1] $ be an arbitrary (noncontinuous) distribution function. Denote with $Q(p)=inf\{x:p \leq F(x)\}$ the associated quantile function. I would like to use that $ \...
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1answer
27 views

Prove that if $f$ is nonnegative, measurable and $E_k \nearrow E$, then $\lim_k \int_{E_k} f = \int_E f$.

Pleas check my proof of the following assertion. Let $\{E_k\}$ be a sequence of measurable sets, and $E_k \nearrow E$. Suppose $f(x)$ is nonnegative and measurable on $E$. Prove $$\int_E f(x) \, dx =...
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1answer
26 views

Counterexample of the converse of Jensen's inequality

Let $\phi$ be a convex function on $(-\infty, \infty)$, $f$ a Lebesgue integrable function over $[0,1]$ and $\phi\circ f$ also integrable over $[0,1]$. Then we have: $$\phi\Big(\int_{0}^{1} f(x)dx\...
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1answer
26 views

Generalize Jensen's Integral Inequality to the product of two functions

Let $E$ be a measurable set with $m(E)>0$. Let $f$, $\gamma$ be two measurable, real-valued function which are finite a.e. on $E$ with $f, \gamma$ and $f\cdot \gamma$ all integrable. Assume $\...
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0answers
20 views

should check the integrand is in $L^{1}$ before changing variables?

I have seen in many lecture notes that spherical coordinates are used to examine the existence of integrals in highr dimensions with radially symmetric integrands. Can one change variables without ...
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1answer
17 views

Convergence of a sequence of measures of sets implies convergence of the indicator functions.

Let $\lambda$ be the lebesgue measure on the Borel sets. Suppose $\lambda(A) < \infty$ Is the following true? $\lambda(B_n) \uparrow \lambda (A), B_n \subseteq A \implies I_{B_n} \to I_A$ a.e. ...
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3answers
35 views

Convergence of Lebesgue integral of an indicator function

Let $f_n(x) = - \frac{1}{n} \mathbb 1_{[0,n]} (x) $ for $n \in \mathbb N$. (The 1 should be the indicator function.) The sequence converges to $-n-1$, right? What about the Lebesgue integral of $...
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1answer
15 views

construction of step functions to show integrability using Beppo Levi's theorem

Let $M\subset\mathbb{R}^n$ be measurable, $f\colon M\to\mathbb{R}$ continuous, bounded. Claim: $f$ is Lebesgue-integrable. I was able to prove it for $M$ additionally bounded. How to reduce the ...
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20 views

How to prove this result somewhat similar to Du Bois-Reymond's Lemma?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, \dots, n$, $f_{ij}$ is a smooth function. If for every $v\in C^...
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1answer
39 views

Prove that if $f^2(x)$ is Lebesgue integrable on E, so is $f(x)$.

Please check my proof, thank you. Let $f(x)$ be a nonnegative and measurable function defined on the set $E$ with $m(E) < \infty$. Prove that if $f^2(x)$ is Lebesgue integrable on E, then ...
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1answer
31 views

Convergence of sequences of periodic functions

Given $f_n(x) = F(x) (\cos(\pi x))^n$, where $n \in \mathbb N , F: \mathbb R \to \mathbb R$ integrable. Simple question: does the Lebesgue integral converge in $\mathbb R$? (and how to show?)
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Can you prove Lusin's theorem without using approximation of $L_1$ functions?

Lusin's theorem states that if $f:[a,b]\rightarrow\mathbb{R}$ is a Lebesgue measurable function, then for any $\epsilon>0$ there exists a compact subset $E$ of $[a,b]$ whose complement has Lebesgue ...
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24 views

Is it possible to prove Fubini’s Theorem without Dynkin’s Theorem or the Monotone Class Theorem?

Fubini’s Theorem for Lebesgue integrals states that if $X$ and $Y$ are Sigma-finite measure spaces then the integral of a (well-behaved) function $f(x,y)$ with respect to the product measure on $X\...
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1answer
34 views

Why is not $\infty$ allowed as a values of Lebesgue integral?

$\infty$ is allowed as a value of Lebesgue measure $m(E)$ and function $f(x)$, but why do not we say $\int_E f= \infty$?
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Prove that $\lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0$. Where $\int$ represents Lebesgue integration.

Please check my proof, thank you. Prove \begin{align*} \lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0 \end{align*} Proof. Let $f_n(x) = x^n$. Since $f_n$ is a polynomial it is continuous and ...
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2answers
20 views

Integral of a product of a ${L^1}_{loc}$ function and a compact support function

I'd like to prove that, if $f\in {L^1}_{loc}(\mathbb{R}^n)$ and $g$ is a bounded, measurable function with compact support and if $\int_{\mathbb{R}^n} fgdx = 0$ $\forall g$, then $f$ is almost ...
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1answer
57 views

Integrable if absolutely summable

Let $P(\mathbb N)$ the power set of $\mathbb N$ and $f$ the counting measure on $(\mathbb N, P(\mathbb N) )$. If $\{ a_n \}_{n \in \mathbb N}$ is a real-valued sequence, then, so the statement in my ...
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0answers
21 views

Integral over a set of measure 0

Let's $A$ be such that $\lambda (A) = 0$ (Lebesgue measure). I want to prove that for every measurable function $f$, $\int_A f(x) \lambda(dx) = 0$ I did the following : $|\int_A f(x) \lambda(dx)| &...
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1answer
30 views

$f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$

I want to show: $f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$ Anyone got any hints how to prove ...
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1answer
36 views

Show that Riemann integral and Lebesgue integral coincide.

I'm proving that for $f: (\Omega, \mathcal{F}) \to [0, \infty]$ and a $\sigma-$finite measure $\mu$ on the $\sigma$-algebra $\mathcal{F}$, we have $$\int_\Omega f d\mu = \int_{0}^\infty \mu \{f \geq ...
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3answers
69 views

Convolution in $L^2(-\pi ,\pi )$

I'm reading about the Fourier transform and it's properties in The Spectral Analysis of Time Series - Koopmans, L. and I found a particular statement that makes me feel a little bit skeptical. Lets ...
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0answers
53 views

How to show that $\int_{(0,1)}\frac{t^{x-1}}{1+t^y}d\lambda(t)=\sum_{n=0}^{\infty}\frac{(-1)^n}{x+ny}$?

Let $x,y \in \mathbb{C}, \Re(x)>0, \Re(y)>0$. How can I show that $t \mapsto \frac{t^{x-1}}{1+t^y}$ is Lebesgue integrable on $(0,1)$? And furthermore I want to show $$\int_{(0,1)}\frac{t^{x-1}...
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1answer
21 views

Showing a product of two Lebesgue integrals is $\geq 1$ if the product of the integrands is $\geq 1$

Let $\mu$ be a probability measure on a set $X$, i.e. $\mu(X)=1$, and let $f$ and $g$ be positive measurable functions on $X$. Show that if $fg\geq1$, then the integral of $f$ times the integral of $...
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1answer
15 views

Integral of positive functions

Given a measurable space $(X,\mathcal{T},\mu)$ and a measurable function $f:X\to\mathbb{R}_+$, the Lebesgue integral of $f$ is usually defined as the least upper bound of the integrals of finite ...
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1answer
17 views

A simple function and its canonical form.

Simple functions are of the form $\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$ where $\chi$ is the indicator function and that $A_k$'s are measurable sets. This is how Stein defines a simple function ...
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28 views

Lebesgue integral of a sequence of floor functions

Let $f:[0,1) \rightarrow [0,1)$ be the identity function $f(x)=x$ for $x \in [0,1)$. For $n \in \mathbb{N}$, let $f_n:[0,1) \rightarrow [0,1)$ by $f_n(x)=\lfloor2^nx\rfloor/2^n$ ; in other words, for $...
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1answer
82 views

Integration/measure theory “paradox”?

I have encountered the following "paradox." Consider a dense countable subset of $\mathbb{R}$, e.g. $\mathbb{Q}$. Because the set is countable we may parametrise it by $\mathbb{Q} = \{ a_n \}_{n=1}^\...
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0answers
33 views

Counterexample of discontinuous function for integration by parts

Let $f$ and $g$ be continuous and monotone increasing functions on $[a,b]$. Then the integration by parts formula for Lebesgue-Stieltjes integrals \begin{equation*} \int_a^b fdg+\int_a^b gdf =f(b)g(b)...
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1answer
28 views

Let $\mu$ be a probability measure on $\mathcal{B}(\mathbb{R}^d)$. Find $\int \mu(I_x)dx$ [closed]

Let $\mu$ be a probability measure on $\mathcal{B}(\mathbb{R}^d)$. Find $\int \mu(I_x)dx$, where $$I_x := [x_1, x_1 + a_1]\times ... \times [x_d, x_d + a_d], a_j > 0$$ I have $\mu(A_1 \times ... ...
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1answer
33 views

Show that $\int_0^\infty \frac{\sin(xt)}{e^t-1}dt = \sum_{n=1}^\infty \frac{x}{n^2+x^2}$

Show that $\int_0^\infty \frac{\sin(xt)}{e^t-1}dt = \sum_{n=1}^\infty \frac{x}{n^2+x^2}$. From an earlier question I have $ \int_o^\infty \frac{x^p}{e^x-1}dx = p!\sum_{n=1}^\infty \frac{1}{n^{p+1}}$ ...
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1answer
22 views

formula for the integral of characteristic functions of measurable sets

Fix $a_1,\dots,a_n\in\mathbb{R}\setminus\{0\}$ and let $L:\mathbb{R}^n\to\mathbb{R}^n$ given by $L(x_1,\dots,x_n)=(x_1/a_1,\dots,x_n/a_n)$. Let $f:\mathbb{R}^n\to\mathbb{R}$ be a funtion such that ...
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1answer
15 views

Is there a notion of an almost measurable function?

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lebesgue measurable if the set $\{x:f(x)<a\}$ is Lebesgue measurable for all real numbers $a$. But my question is, what if something slightly ...
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1answer
32 views

Lebesgue integrability of $\frac{f(x)}{x}$

Let $f$ be Lebesgue integrable, $f(0)=0$ and assume that $f'(0)$ exists. Show that $\frac{f(x)}{x}$ is Lebesgue integrable. I don't know what to do. The claim makes sense to me because the only ...
2
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1answer
38 views

Prove that the measure of the image of G-delta set, $m(f(G))$, is the integral of the derivative, $\int_{G}f^{'} dm$

Consider a strictly increasing, absolutely continuous function $f: [0,1] \to \mathbb{R}$. Prove that for any $G_{\delta}$ set $G \subseteq [0,1]$, that the Lebesgue measure $m$ of $f(G)$ is given by ...