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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41 views

A question on Lebesgue integral

Let $f\in L^1(\mathbb{R}^n)$. I want to prove that when $m(B-B’)\to 0$ $$\left|\int_{B’}f(x)dx-\int_{B}f(x)dx\right|\to 0.$$ In Riemann integral, if $f$ is absolutely integrable on $[a,b]$, then $|f|$...
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1answer
18 views

Substitution/Transformation to integrate $(1+\langle x,x\rangle)^{-2}$ over $\mathbb{R}^2$

I want to integrate $f:=(1+\langle x,x\rangle)^{-2}$ over $\mathbb{R}^2$. Geometrically this problem is finding the volume of the stack of disks $$\frac{1}{\sqrt{z}}-1\ge x^2+y^2$$ I'm sure there is ...
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2answers
35 views

Find Lebesgue Integral: $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$

Consider a sequence of functions $f_n:[0,2]\rightarrow\mathbb{R}$ such that $f(0)=0$ and $f(x)=\frac{\sin(x^n)}{x^n}$ for all other $x$. Find $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$. My ...
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1answer
39 views

Proof of dominated convergence theorem

I was going through the proof of the Dominated Convergence Theorem. Now if we have that ($f$$_n$) is a sequence of measurable functions such that $\lvert f_n\rvert$ $\le$ $g$ for all n where g is ...
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1answer
48 views

Show there exists $\gamma$ such that $\int f|g|=\gamma \int |g|$

I have been going through a course in measure theory and integration. Got stuck in one intermediate step of the proof.It says that- If $g$ is integrable on $\mathbb{R}$ and $f$ is measurable and ...
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2answers
80 views

Does $\{f_n\}$ converges to $f$ in $L_2$?

Definition We say that a sequence $\{f_n\}$ converges quadratically to $f$ in $L_2$ when $$lim_{n \rightarrow \infty} \int_X[f_n(x)-f(x)]^2d\mu(x)=0$$ Let $f_n:[0,1] \rightarrow \mathbb{R}$ ($n=1,2,.....
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1answer
75 views

Measures=distributions, what is wrong?

We know that in $\mathbb{R}^N$ distributions can be defined as $D(\mathbb{R}^N)=C_c^\infty(\mathbb{R}^N)'$ and Radon finites measures as $m(\mathbb{R}^N)=C_0(\mathbb{R}^N)'$. So, from this definitions ...
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1answer
42 views

Derivative of $F(x)=x^2\sin(1/x^2)$ exists for all x, but fails to be (Lebesgue) integrable.

Consider the function $F(x)=x^2\sin(1/x^2)$ for $x\not=0$, and $F(0)=0$. Show that $F^{'}(x)$ exists for every x, but that $F^{'}(x)$ is not integrable on [-1,1]. So the part I am struggling with ...
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0answers
11 views

Fundamental theorem of line integral for Lipschitz continuous function

Suppose that $f : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continuous, can we conclude that $$ \int_{[0,1]} (y-x)^T \nabla f(\mu y + (1-\mu )x) \text{d}\mu = f(y) - f(x) $$ where $\int$ stands for ...
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106 views

Is Jost's version of the Lebesgue integral well-defined?

In Postmodern Analysis by Jurgen Jost, the Lebesgue integral of a step function is defined as follows: Suppose we have a step function $t:W\subset\mathbb{R}^d\to \mathbb{R}$ defined on a cube $W\...
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1answer
40 views

Convergence of expectation of inverse sum of random variables

Suppose $X_1,..,X_n,...$ are positive random variables, such that $(X_1+...+X_n)/n\to\mu$ almost surely as $n\to\infty$. I intuitively believe that $\mathbb{E}\Big[\frac{Y}{(X_1+...+X_n)/n}\Big]\to\...
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51 views

Radon-Nikodym Theorem (Folland 3.8)

I am reading the proof of Radon-Nikodym Theorem in Follands "Real Analysis Modern Techniques and Their Applications". Specifically, Theorem 3.8 on page 90. Folland starts with the case that both the ...
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1answer
30 views

How to prove it is a continuous function?

Assume that $f\in L^1(\mathbb{R}^n)$. Let $B(x,r)$ be a ball centered at $x$ with radius $r$. By Lebesgue Differentiation Theorem, $$ \lim_{r\to 0} \dfrac{1}{m(B(x,r))}\int_{B(x,r)}f(y)dy=f(x)$$ ...
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22 views

Evaluate $\iint_{A\times B}\mathbf{1}_{E}dxdy$

I'm trying to prove that partial and multiple integrals of $\mathbf{1}_{E}$ are equal, being $\mathbf{1}_{E}$ a two-dimensional indicator function. To be said, i'm trying to prove that $$\iint_{A\...
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71 views

Prove $f$ is constant almost everywhere in $[a,b]$ [closed]

Suppose $f$ is Lebesgue integrable. If for any $g$ is Lebesgue integrable s.t. $\int_a^bg(x) dx = 0$, we have $$ \int_a^bf(x)g(x)dx = 0 $$ Prove $f$ is constant almost everywhere in $[a,b]$
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1answer
30 views

Is there an integrable g that dominates all fn's, $f_n(x)=\frac{\chi_{[0,n]}(x)}{n^p}$?

Is there an $ g\in L_1([0,\infty]) $that dominates all $f_n$'s, $f_n(x)=\frac{\chi_{[0,n]}(x)}{n^p}$, $n\geq 1, 0 < p <\infty$? If exists how can I find it? I did "a proof" for p=1 there is $x\...
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1answer
32 views

Bochner integral in a direct sum of Banach spaces

Let $\mathcal{B} = \mathcal{B}_1\oplus\ldots\oplus \mathcal{B}_n$ be a direct sum of Banach spaces $\mathcal{B}_i$ each with norm $\|\cdot\|_{\mathcal{B}_i}$. The Banach space $\mathcal{B}$ has many ...
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1answer
42 views

Notation on Measure Theory and Integration

What is the meaning of $L^{2}_{loc}(\mathbb{R}, L^{2}(\Omega))$?, where $\Omega\subset\mathbb{R}^{d}, d\geq 1; L^{2}(\Omega)\equiv \mathcal{L}^{2}(\Omega)/ker(\|\cdot\|_{2})$ Is there some good ...
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19 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
36 views

Value of the function at such points are finite or not

This is Theorem 1.3 in Stein’s Real Analysis. If $f$ is integrable on $\mathbb{R}^d$, then $$\lim_{\substack{m(B)\to 0\\ x\in B}}\dfrac{1}{m(B)}\int_B f(y)dy=f(x)~~~{\rm for~a.e.~}x\in \mathbb{R}...
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16 views

Quantitative Lebesgue differentiation theorem with strong regularity lemma

I am reading and studying one article in Tao's blog. Several quantitative versions of Lebesgue differentiation theorem are discussed there. I have gone through most of the proofs he left to the ...
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1answer
43 views

How to find closest trigonometric polynomial?

Could you give me any hints? How to find the trigonometric polynomial of the third order closest to the function: $f(x) \in L_{2\pi}^2, \,f(x) = \text{sgn}(x), \quad x \in (-\pi, \pi)$ in the norm $...
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1answer
44 views

A question relate to dominated convergence theorem

given that:$g(x)=\frac{1}{xln(x)},x > 1$. $f_n=c_n\mathbf{1}_{A_n}$, where $c_n \geq 0$, $A_n$ is measurable and $\subset [2, +\infty)$. $|f_n|\leq g$ ($f_n$ is dominated by $g$.) $\forall x, \lim_{...
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2answers
53 views

Compute $\lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin{(x/n)} \sin{(e^x)}dx$.

Another practice preliminary question for you all. This time, a double limit of an integral. Problem Compute $\lim_{n \rightarrow \infty} \lim_{R \rightarrow \infty} \int_0^R \sin{(x/n)} \sin{(e^...
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1answer
40 views

Integrability of $\frac{x_1}{|x|^{n}}$ over the unit ball

Is $\frac{x_1}{|x|^{n}}$ integrable over the unit ball of $B_1(\mathbb{R}^n)$? That is, is $$\int_{B_1(\mathbb{R}^n)} \frac{|x_1|}{|x|^{n}}<\infty?$$ I know that $|x|^{-a}$ is integrable over the ...
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1answer
26 views

Prove that $\int\sum k\chi_{f^{-1}}<\infty$

For a Lebesgue measurable sets and functions problem, I need to prove this statement: Being $A\subset\mathbb{R}$ a measurable set with $m(A)<\infty$, and $f:A\rightarrow[0,\infty)$ a Lebesgue ...
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1answer
16 views

Does null average against every smooth function implies independence?

Are these assertions equivalent? $f:\mathbb{S}^1\times \mathbb{S}^1\to\mathbb{C}$ is such that $$ \int_0^{2\pi}\int_0^{2\pi}f(x,y)\psi(y)dydx=0$$ for all $\psi\in C^{\infty}(\mathbb{S}^1).$ $f:\...
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0answers
84 views

A problem with Lebesgue integral in functional analysis

Let $ F \in D '$ have a compact support, $ F(\varphi)\geq0$ for any $\varphi \geq0$. How can i prove that $F(\varphi) = \int \! \varphi \, \mathrm{d}\mu$ for some non-negative measure $ \mu $?
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1answer
25 views

$f, g$ are probability density functions of an exponential distribution, prove h is $\gamma (\lambda ,2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. Then is asked: $f, g$ are probability density functions of an ...
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1answer
45 views

I need to evaluate $\iint_S xdydz + (x+y)dz dx+(x^2+2z)dxdy$

My problem asks me to evaluate the integral (using direct integration) $$\iint_S (x)dy\wedge dz + (x+y)dz\wedge dx+(x^2+2z)dx\wedge dy$$ Being $S$ the surface of the solid limited by the following ...
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0answers
27 views

If $f$ is Lebesgue integrable, then show that $h(u)=\int_{\mathbb{R}} e^{iux} f(x) dx$ , for real $u$.

Suppose we have a function $h$ which is continuous and Lebesgue integrable on $\mathbb{R}$. We have $f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-iux} h(u) du $, for all real $x$. If $f$ is Lebesgue ...
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1answer
23 views

Inequality on integrals of $L^1$ functions

Let $\lambda \geq 0$ and $(X,d,\mu)$ be a $\sigma-$finite measure space. Then for $f, g \in L^1(X,\mu)$ $$ \left| \int_X (|f|-\lambda)^{+} d\mu - \int_X (|g|-\lambda)^{+} d\mu \right| \leq \int_X ||f|...
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32 views

How does this inequality follow? Fatou's Lemma and DCT

I am reading this answer, and I am not sure how we get $$ \int g-\int f\leq \int g-\limsup\int f_n. $$ I see that the integral can distribute over the $-$ on the LHS, but I am not seeing how the $\...
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1answer
22 views

Computing the limit of a sequence of functions fn : R → R

Consider a sequence of functions $f_n : (0,\infty) \rightarrow \Bbb R$ defined by $$f_n(x)=\frac{n}{n + x + nx^2}$$ Show that $f_n(x)\le f_{n+1}(x)$ for all $n \in \Bbb N$ and $x \in (0,\infty)$. ...
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67 views

Is this function lebesgue integrable or not?

I'm trying to see if this function is lebesgue integrable. $$\int_0^1 \frac{(-1)^{\lfloor 1/x \rfloor}}{x^2} dx.$$ How can I prove it? I try the following: Let $f(x)=\frac{(-1)^{\lfloor 1/x \rfloor}...
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1answer
57 views

Prove $\lim_{n \rightarrow \infty} f(x) f(2^2x) f(3^2x) \cdots f(n^2x) = 0$ for $f: \mathbb{R} \rightarrow \mathbb{R}$ in $L^1(\mathbb{R})$.

Here's another question that I'm stuck on from my studies for an upcoming exam. This one comes from another practice preliminary exam. Problem Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a ...
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59 views

I need to prove $\int\sum a_k=\sum\int a_k$ for a certain $\{a_n\}$

I've been solving a problem from my integral calculus class and I've found I need to prove that: $$\int^\infty_{-\infty}\sum^\infty_{k=0}((-1)^k\frac{(ax)^{2k}}{(2k)!}e^{-x^2})dx=\sum^\infty_{k=0}((-1)...
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0answers
29 views

Lebesgue's fundamental theorem of calculus

The second part of the fundamental theorem of calculus states that If $f:[a,b]\to\mathbb{R}$ is a Lebesgue-Integrable function and $F$ is a primitive of $f$, then $\int_{a}^{b}f(x)dx=F(b)-F(a)$. I ...
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1answer
40 views

Lp space Example

How are spaces connected $L_{\infty}(E)$ and $L_{p}(E)$, $|E| = \infty$? $$f \in L_\infty, \text{ but } f\notin L_1 \quad f = \frac{1}{x} \quad E = [1, \infty)$$ What can we say about reverse ...
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1answer
18 views

A sufficient condition for a Lebesgue point

Let $f\in L^1(\Bbb R^n)$ and let $x\in \Bbb R^n$. $x$ is said to be a Lebesgue point of $f$ if $\lim_{r\to 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f(y)-f(x)|~dm(y)=0$ where $m$ is Lebesgue measure on $\...
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1answer
109 views

For measurable $f: \mathbb{R} \rightarrow \mathbb{R}$ prove $f(x)$ and $\frac{1}{f(1/x)}$ cannot both be Lebesgue integrable.

First question on MSE! I'd appreciate hints, theorem suggestions, or method suggestions regarding the question in the title or below. Please avoid full solutions. I'm studying for an exam coming up ...
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1answer
93 views

If $g\in L^1[0,1]$ satisfies $\int_{[0,1]} fg~dx=0$ for all $f\in C[0,1]$ then $g=0$

Suppose $g\in L^1[0,1]$ satisfies $\int_{[0,1]} fg~dx=0$ for all $f\in C[0,1]$. (Here, we are considering Lebesgue measure on $[0,1]$. ) Then do we necessarily have $g=0$? I am trying to show that "...
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0answers
6 views

Prove $\sigma$-sublinearity of Lebesgue upper-integral

I'm having trouble understanding the proof of the $\sigma$-sublinearity of the Lebesgue upper-integral given in my notes. The property: Let $f,g:\mathbb{R}^d \to [0,+\infty[$ and $c\ge 0\,\,(c\in\...
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1answer
102 views

Integrability of $\frac{1}{(x^2+y^2+z^2)^a}$ on $E=\{(x,y,z)\in \mathbb{R}^3: z>1, \ z^2(x^2+y^2)<1 \}$

Let $E=\{(x,y,z)\in \mathbb{R}^3: z>1, \ z^2(x^2+y^2)<1 \}$ and $$f_{a}(x)=\frac{1}{(x^2+y^2+z^2)^a}$$ I need to find all $a\in \mathbb{R}$ such that $f_a\in L^1(E).$ I already know a ...
3
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1answer
145 views

Evaluate the integral $\iint_S(x)dy\wedge dz+(x+y)dz\wedge dx+(x^2+2z)dx\wedge dy$

In a problem from my multivariable integration class, i've reached this problem. I will thank any comment with advice or answer. The problem asks me to calculate the integral $$\iint_S(x)dy\wedge dz+(...
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1answer
44 views

Pass to the limit under the sign of the integral of $f_{n}(x)$ [closed]

I'm trying to investigate following limit: $$ \lim_{n \to \infty}\int_{0}^{\frac{\pi}{2}}\frac{cos^{n}x}{1 + x^{3}}dx $$ I have a couple of questions 1. Is it possible to find the limit by the ...
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2answers
99 views

$(\int f_1d\mu)^2+\cdots+(\int f_nd\mu)^2\leq(\int \sqrt{f_1^2+\cdots+f_n^2}d\mu)^2$

Let $(X, \mathfrak{B}, \mu)$ be a measurable space, possibly not $\sigma$-finite, and $f_1, \cdots, f_n \colon X\to (-\infty, +\infty)$ be integrable functions on $X$. Does $$(\int f_1d\mu)^2+\cdots+(\...
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1answer
40 views

Help to evaluate the integral $\iint_D\frac{y}{\sqrt{x^2+y^2}}dxdy$

I'm solving a problem about integrals in curves, and I got this integral: $$\int_1^2\int_1^2\frac{y}{\sqrt{x^2+y^2}}dxdy.$$ I have been struggling to solve it. I'm sure i have to do some variable ...
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0answers
11 views

Expected value of random variable over a shrinking set. (Left-derivative of superexpectation)

I am working on a proof related to the left-derivative of the superexpectation operator $E_X(x) = E[\max\{X, x\}]$. Let $X \in L^2(\Omega, \mathcal{F}, P)$ be a random variable. Let $x', x \in \...
4
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1answer
48 views

Is $(X, Y)$ always absolutely continuous with respect to $P_X \otimes P_Y$?

Definitions: Let $X: (\Omega, \mathcal A) \to (\mathbb R, \mathcal B)$ be a random variable on the probability space $(\Omega, \mathcal A, P)$ and define its distribution as the probability measure $...