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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10 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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42 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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21 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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23 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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21 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
20 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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30 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
21 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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61 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
53 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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53 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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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
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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
16 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
96 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
86 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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If $f\ge 0$ is Riemann-integrable, then the Lebesgue upper-integral of $f$ is equal to the Riemann-integral of $f$.

Property: If $f\ge 0$ is Riemann-integrable, then $(L)\overline{\int} f = (R)\int f$. Consider following lemma, where $\mathcal{T}$ is the set of step functions, $\mathcal{C_c}$ the set of ...
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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
83 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 ...
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1answer
119 views
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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
41 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
94 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
38 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 \...
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1answer
47 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 $...
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1answer
50 views

How calculate ${\lim}_{n→∞}\int_0^{\pi/2}{7^{-2^n\cos^nx}dx}$ [closed]

How calculate it? $$\lim_\limits{n\to\infty}\int_0^{\pi/2}{7^{-2^n\cos^nx}dx}$$
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1answer
54 views
+50

Can we get simultaneous convergence of the integral using simple functions?

Let $Y$ be an integrable nonnegative random random variable on some probability space $X$, and Let $F:[0,\infty) \to \mathbb R$ be a continuous function. Suppose that $F(Y) \in L^1(X)$. (I am fine ...
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1answer
26 views

Lebesgue measure/integral problem - where do I go wrong?

Let $f\geq0$ be a bounded function supported in a measurable set $E$ with $m(E)<\infty$. Show that if $\int_E f=0$, then $m(E)=0$. My proof: Let $E_\epsilon=\{x\in E:f(x)\geq\epsilon\}$. Then for ...
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0answers
26 views

Is it possible to write this improper Riemann integral as a proper Lebesgue integral?

Let: $V' \subset \mathbb{R^3}$ $S' = \partial V' \subset V' \subset \mathbb{R^3}$ $\mathbf{r}=$ position vector of field point $\mathbf{r'}=$ position vector of source point $\...
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1answer
60 views

Find a solution [closed]

If $g(x) = \int_0^x f(z) \,dz$ , $,$ and $\int_0^1 F(z) \,dz = 31$, then find $\int_0^1 f(z)\, dz$. I got stuck in this question. Can someone help me?
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2answers
73 views

Calculate the volume of the solid determined by $S_1$ and $S_2$

I want to calculate the volume of the solid determined by this tho surfaces: $$S_1=\{(x,y,z)\in\mathbb{R}:x^2+y^2+z^2=R^2\}$$ $$S_2=\{(x,y,z)\in\mathbb{R}:x^2+y^2=Rx\}$$ The solid is the intersection ...
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1answer
111 views

Evaluate the integral $\int_\Gamma(y^2-z^2)dx+(z^2-x^2)dy+(x^2-y^2)dz$

I need to calculate the integral $$\int_\Gamma(y^2-z^2)dx+(z^2-x^2)dy+(x^2-y^2)dz$$ being $\Gamma=S_1\cap S_2$, given: $S_1=\{(x,y,z)\in\mathbb{R}:2x+2y+z=3\}$ $S_2=\{(x,y,z)\in\mathbb{R}:z=9-x^2-y^2\...
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16 views

Question regarding definition of Lebesgue integral

I am reading Measure Theory by Halmos, and I was wondering if someone could help me on the following definition: If there exists a mean fundamental sequence $\left\{f_{n}\right\}$ of integrable ...
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22 views

Why does limit of lebesgue integral of a sequence of simple function exist? [closed]

I have a question regarding the definition of lebesgue integral. As shown in the image, integral of f is defined as lim of integral of fn. I was wondering why does this limit exists? (In the ...
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24 views

Dominated convergence theorem for $\lim_{r\to 1^-} \int_0^{2\pi} \text{Log} \left|1-re^{it}\right| \, dt$

I am trying to apply the DCT to see that $$\lim_{r\to 1^-} \int_0^{2\pi} \text{Log} \left|1-re^{it}\right| \, dt $$ exists and is finite, where $\text{Log}$ is the main branch of the logarithm. It is ...
2
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2answers
40 views

Lebesgue integral, Is the solution right?

I'am trying to understand Lebesgue integration Compute $\int_{0}^{\pi}$ f(x)dx Where $f(x) = \begin{cases} sin x & \text{ if } x \in \mathbb{I} \\ cosx & \text{ if } x \in \...
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2answers
28 views

Is this version of the dominated convergence theorem true?

Let $I$ be a real interval, $t_0 \in I,$ $E$ a measurable set in $\mathbb{R}$ and $f \colon I \times E \to \mathbb{R}$ a function such that: $f(t, \cdot) \in L^1(E)$ for every $t \in I;$ There exists ...
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0answers
15 views

Show that $u(x,t)=\int_{\mathbb{R}^n}{\exp\left(-\frac{|x-y|^2}{2t}\right)f(y)dy}$ is $C^\infty$ and converges locally uniformly

Let $f$ bounded continuous function in $\mathbb{R}^n$ then the function $$u(x,t)=\int_{\mathbb{R}^n}{\exp\left(-\frac{|x-y|^2}{2t}\right)f(y)dy}$$ Is $C^{\infty}-$smooth in $\mathbb{R}^n\times\...
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0answers
35 views

Set of lower dimension has Lebesgue measure zero?

Let $f(x):\mathbb{R}\to\mathbb{R}$ be a continuous strictly monotonic function. For example, $f(x)=\tanh(x)$. Does the set $\{\mathbf{x} \in \mathbb{R}^n : \sum_i f(x_i) = 0 \}$ have Lebesgue measure ...
2
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1answer
38 views

Monotonicity of the Lebesgue Integral

I am working through Baby Rudin, and encountered the following remark: If $f$ and $g\in\mathcal{L}(\mu)$ on $E$, and if $f(x)\leq g(x)$ for $x\in E$, then $$\int_{E}fd\mu\leq \int_E gd\mu.$$ ...
2
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2answers
76 views

$f(x)=1/(|x|+1)^k$ defined on $\Bbb R^k$ is not integrable

Consider the function $f(x)=1/(|x|+1)^k$ defined on $\Bbb R^k$. I am trying to show that $f$ is "not" integrable on $\Bbb R^k$, i.e., $\int_{\Bbb R^k}f~dx=\infty$. I tried to show that $\int_{B(0,r)...
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0answers
51 views

Convergence of integral by using a Lebesgue point

I need some help with this problem: let $\lambda>0$ and $w(x)=Ce^{-|x|}$ a function defined on $\mathbb{R}^{N}$, where $C>0$ is a constant such that $\int_{\mathbb{R}^{N}}w(x)dx=1$. Then we ...
0
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1answer
70 views
+50

Prove that $f\in L^1(A)\Leftrightarrow \sum_{n}^{\infty}m(\{ x\in A : f(x)\geq n \}) < \infty$

I'm stuck with some problem of my Integral Calculation in Several Variables course. The problem goes like this: Let $A\subset \mathbb{R}$ be a measurable set with $m(A)<\infty$, and $f:A\...
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27 views

Simple Change of Variables in Lebesgue Integration

I'm not sure if these details matter, but anyway for this particular case, consider a compact abelian group $G$ with operation $\cdot$, a Haar measure $\mu$ on it and $f$ a non-trivial character on $G$...
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1answer
173 views

For $g(x) = 1/x$ extended to complex-values, what is antiderivative of $g$?

My question is similar to the questions For $g(x) = 1/x$, determine the antiderivative, and determine its definite integral. Except I want to use complex-valued variables and functions. First, ...
1
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1answer
32 views

For (Lebesgue) measurable functions $f$ and $g$, if $f=g$ a.e., then $ \int_{E} f=\int_{E} g. $

Problem Let $f$ and $g$ be bounded (Lebesgue) measurable functions defined on a set $E$ of finite measure. If $f=g$ a.e., then $$ \int_{E} f=\int_{E} g. $$ Attempt Let $f,g:E\to \mathbb{R}$ ...
2
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2answers
45 views

Measure theory: motivation behind monotone convergence theorem

I am watching a very nice set of videos on measure theory, which are great. But I am not clear on what the motivation is behind the monotone convergence theorem--meaning why we need it? The ...
1
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1answer
77 views

Deep doubt on a double surface integral

I don't understand how to proceed with an exercise. I will write down what I have done so far. The exercise is: Evaluate the following integral $$\iint_{\Sigma}\dfrac{1}{x^2+y^2}\ \text{d}\sigma $$...
2
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1answer
33 views

Give an example function $f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$ [closed]

Give an example function $f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$
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1answer
22 views

Multiplication of simple function looks like?

I was wondering what does the multiplication of two Lebesgue integrable simple functions look like. Assume integral of a function f is defined as

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