For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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10 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
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1answer
14 views

calculate $\lim_{n\to\infty}\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$

We've had the following Lebesgue-integral given: $$\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$$ How can you show the convergence for $n\rightarrow\infty$? We've tried to use ...
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1answer
45 views

If $\int_0^{x} g \leq \int_0^x f$ and $\phi$ is nonincreasing then $\int_0^{\infty} \phi g \leq \int_0^\infty \phi f$

Let $f, g$ be measurable real-valued functions on $[0, \infty)$, with $$\int_0^{x} g \leq \int_0^x f$$ for each $x$. Show that if $\phi: [0, \infty) \rightarrow [0, \infty)$ is nonincreasing, then ...
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1answer
36 views

If $\int_0^1 f(y)\sin(xy) dy = 0$ for every $x$, then $f = 0$ almost everywhere.

Can someone please give me a hint on this question, I have no idea where to start. Let $f \in L^p$ for some $1 \leq \infty$. Assume for all $x \in [0,1]$ that $$\int_0^1 f(y)\sin(xy) dy = 0$$ Show ...
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1answer
18 views

Prove $\lim_{n \rightarrow \infty} n \cdot \lambda(\{ f > n \}) = 0$. Where $\lambda$ is the lebesgue measure.

Suppose $f$ is integrable over $E$. And the assumptions are as given above. Then, currently, I have from Chebyshev's inequality, $$ \lambda( \{ f > n \} ) \leq \frac{1}{n}\int_E |f| $$ Thus, ...
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1answer
21 views

example of a sequence $f_n$, $n=1,2…$ of integrable functions converging to $f$ s.t limit of integral of $f_n$ does not exist

Is there an example of a sequence of of functions $f_n$ converging to a function $f$ such that $f_n$, $n=1,2...$ are integrable and nonegative and their integral over a measurable set $A$ is less than ...
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0answers
34 views

Lebesgue integration of $f(x)=\frac{1}{x}$ where $x\in[0,3]$

We have the function $f(x)=\infty$ if $x=0$ and $f(x)=\frac{1}{x}$ if $x$ otherwise. So, in this two values of function, I made simple approximation of $f(x)$ by the help of simple function : ...
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1answer
21 views

Convergence in L^p, Cauchy in L infinity

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$? Is it true if $u_n$ are all test functions, ...
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1answer
48 views

$f \in L^1$, but $f \not\in L^p$ for all $p > 1$

"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$." I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
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0answers
44 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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0answers
10 views

Measurability of $f:X\times Y\to\mathbb{K}$ and $f(-,y):X\to\mathbb{K}$

Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of ...
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1answer
16 views

$ \int_{\mathbb{R}^n} f^p dx$ for $p>0$ and measurable $f$

Let $f: \mathbb{R}^n \rightarrow \mathbb{\overline{R}} $ be non-negative and (Borel)-measurable and $p>0$. Then: $$ \int_{\mathbb{R}^n} f^p dx = p \int_{0}^{\infty} t^{p-1} ...
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1answer
27 views

Uniform convergence and integrability

If $(f_n)_{n \in \Bbb N}$ converges to $f$ uniformly and each $f_n$ integrable would it imply $f$ is integrable and $$\lim_{n \to \infty}\int f_n = \int f$$ In case each $f_n$ is nonnegative ...
4
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1answer
50 views

Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
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1answer
27 views

For which $s$ is the function $(||x||^{s-2}x_i)^2$ integrable on the unit ball of $\mathbb R^n$?

Initial task is to find out, for which $s$ stands $u=||x||^s \in H^1(\Omega)$, where $\Omega = B(1,0)\subset\mathbb{R}^n$ and $H^1(\Omega)$ is a Sobolev space $W^{1,2}(\Omega)$. As to prove this, we ...
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1answer
38 views

Lebesgue point and integration

Let $f$ be in $L^1_{\text{loc}}(\mathbb{R})$. We know that for almost every $t$ $$ \lim_{h\to 0} \frac{1}{h} \int_t^{t+h} |f(u)-f(t)|\text{d} u = 0. $$ My question is : can we say that for almost ...
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0answers
33 views

Absolute continuity as a condition for $F[f^{(k)}](\lambda)=(i\lambda)^k F[f](\lambda)$

In read in Kolmogorov-Fomin's (p. 429 here) that if function $f$ is such that $f^{(k-1)}$ is absolutely continuous on any interval and if $f,...,f^{(k)}\in L_1(-\infty,\infty)$, [...] we get ...
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1answer
22 views

If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: ...
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2answers
52 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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1answer
59 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
3
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1answer
44 views

Show that Fourier transformation is differentiable if $\int|xf(x)|\,d\lambda<\infty$

Let $f\in\mathcal{L}^1(\mathbb{R},\mathcal{M},\lambda)$. Then we define the Fourier transform of $f$, denoted $\hat{f}$, by ...
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1answer
27 views

Prove that a subset is measurable is and only if the measurable of the set equal to the sum of that subset and its complement

Let $X$ be a set and $\mathscr{A}$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a measure on $\mathscr{A}$ such that $l(X) < \infty$. Define $\mu^{*} $ as $$ ...
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1answer
44 views

Proof of lebesgue integral of $f(x)=\frac{1}{x}$ in the interval [1,5] equals to $\ln5-\ln1$

Would everyone please help me on how to prove this value of Lebesgue integral of the function $f(x)=\frac{1}{x}$ in the interval [1,5] by using approximation by simple function $f_n$ step by step? ...
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2answers
39 views

Let $f$ be measurable and $a,b\in\mathbb{R}$ with $\frac{1}{\lambda(M)}\int_Mf\ d\lambda \in [a,b]$ Show that: $f(x) \in [a,b]$ almost everywhere.

Assignment: Let $f$ be Lebesgue - measurable and $a,b \in \mathbb{R}$ with the property: $$\frac{1}{\lambda(M)} \cdot \int_Mf\ d\lambda \in [a,b]$$ for all Lebesgue - measurable sets $M ...
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2answers
13 views

For non-negative functions, are Riemann-Stieltjes and Lebesgue integrals equivalent?

For functions $f : \Bbb{R} \mapsto \Bbb{R^+ \cup 0}$ which are non-negative everywhere, does existence of the are the Riemann-Stieltjes integral imply existence of the and the Lebesgue integral (and ...
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1answer
21 views

How to show that $\lim_{h\to 0}\int_0^h|f(x)|dx=0.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a locally integrable function. How can we see that $$\lim_{h\to 0}\int_0^h|f(x)|dx=0.$$ If $f$ is bounded, then we have the result. But what about $f$ only ...
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1answer
25 views

$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$ absolutely continuous and summable on $\mathbb{R}$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$. I read that the function$$\varphi(x)=\int_{[\xi_0,\xi]}f(x+t)d\mu_t$$is absolutely continuous on any real closed ...
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1answer
65 views

Comparison between lebesgue integral and riemann integral of $f(x)=x^2$ in $[0,2]$

If we have an example $f(x)=x^2$ let's say for $[0,2]$. In lebesgue integral, I already use a sequence of function $f_n(x)$ as approximation to $f(x)$ ($f_n(x)$ converges to $f(x)$) which is stated ...
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2answers
217 views

A Challenge on One Integral Problem

I want to show that if $$I(t)=\int_0^\infty e^{-t^2x}\,\frac{\sinh(2tx)}{\sinh(x)}\,dx$$ then for $t^2\neq1$ $$I(t)=4t\sum_{n=0}^\infty \frac{1}{\left(2n+1+t^2\right)^2-4t^2}$$ Finally, show that ...
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1answer
27 views

If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite

If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite Previous question asked you to prove Markovs inequality so I think it may have ...
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1answer
15 views

For what values of $a > 0$ and $b \in \mathbb{R}$ is the following function integrable?

For what values of $\alpha > 0$ and $\beta\in\mathbb{R}$ is the $$ f:(1,+\infty) \to \mathbb{R}: f(x) = \frac{\arctan(x^\alpha)}{x^\beta}$$ function integrable ? I know that the above, an ...
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2answers
43 views

Proofs of properties of a measureable and Lebesgue integrable function

Could I get some help showing the following properties to be true: a) $f: X \to [0,\infty) $ is measurable and $\int f d\mu < \infty$ $\forall a > 0$, let $X_a = \{x \in X :f(x) >a\}$, show ...
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1answer
51 views

Computing moments

given $\int_{-\infty}^{+\infty} \! e^{-tx^2} \, \mathrm{d}\lambda x = \sqrt{\pi/ t} $ I have been asked to compute the moments $\int_{-\infty}^{+\infty} \! x^{2n} e^{-x^2} \, \mathrm{d}\lambda x $ ...
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0answers
22 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
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1answer
55 views

Integration of $\exp[f(x,y)]$

Here is the question i want to solve. $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp\left[{-2\over3}(y^2-yz+z^2)\right]\,dy\,dz$$ I know that $\exp$ is $e^{f(x)}$ and i can find $\int ...
2
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1answer
30 views

$f$ absolutely contininuous $\Rightarrow f\cdot\sin$ absolutely continuous?

I wonder whether, if $f:[a,b]\to\mathbb{C}$ is an absolutely continuous function, multiplying it by $\cos\frac{2\pi nx}{b-a}$ or $\sin\frac{2\pi nx}{b-a}$ results in another absolutely continuous ...
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1answer
21 views

Two notions of absolute continuity

If ν is a signed measure and µ a positive measure, we say that ν is absolutely continuous w.r.t. µ if µ(E) = 0 ⇒ ν(E) = 0. If |ν| is a finite measure then this ...
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1answer
29 views

Let $f \in L^1$ with $f$ differentiable at zero and $f(0)=0$. Show $\int_{-\infty}^{\infty} \frac{f(x)}{x} dx$ exists.

Is this proof good? Given the problem as stated. I first define, $$ g(x,b) = \frac{f(x)}{x}e^{ibx} $$ Which has the following property, $$ g_b(x,b) = if(x)e^{ibx} $$ And that, $$ |g_b(x,b)| = ...
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0answers
50 views

Prove g is not integrable on any interval

Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let $r_k$, k=1,2,3...be an enumeration of all rationals and set $g(x)=\sum_{k=1}^{\infty}2^{-k}f(x-r_k)$ Prove $g^2$ is finite almost ...
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0answers
15 views

Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.)

Can we have $|\int _{\{f _n < f \} } (f _n - f )d \mu|<|\int _X (f _n - f )^- d \mu | $ where $f _n \to f $ (a.e.) For me these two integrals are identical, but I have proof where the ...
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1answer
20 views

Continuously differentiable functions dense in $L^2[a,b]$

I read in Kolmgorov-Fomin's Элементы теории функций и функционального анализа (p. 408 here) that the set of continuously differentiable functions are dense everywhere in space $L^1[a,b]$ of Lebesgue ...
2
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0answers
28 views

Are these functions Lebesgue integrable?

let's consider the function $$f: [0,1] \to \mathbb{R}^+, \quad f(x) = \begin{cases} x^{-a} & x \in \mathbb{Q} \; \text{and} \; x>0\\ 0 & \text{otherwise}. \end{cases}$$ for some $a \geq ...
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3answers
49 views

Let $f \in L^1$ then prove $\lim_{b \rightarrow \infty} \int_b^{\infty} f(x) dx=0$.

So the question is as stated in the title. We are given the hint to use LDCT. Since this is homework I'm not looking for an explicit solution. I just need hints. For example, my first thoughts were ...
2
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1answer
25 views

Approximation of $f\in C[a,b]$ by functions constant on intervals of length $(b-a)/2^n$

I read (p. 405 here) that a continuous function on $[a,b]$ can be arbitrarily approximated according to the distance of space $L^2[a,b]$ defined by $d(f,g):=\|f-g\|_2=\int_{[a,b]}|f-g|^2d\mu$ with ...
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1answer
29 views

Convergence test for improper multiple integral

I have a function $f:\mathbb R^n \to \mathbb R$ such that $f(x)=(1+|x|)^me^{-\frac{|x|^2}{a}}$. I need to check is $$\int\limits_{\mathbb R^n}f(x)dx = \int\limits_{\mathbb R^n} ...
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2answers
24 views

Prove why this equality holds

Help in a problem about Lebesgue integration inequality If the sum is finite there is no problem , but if it is not how i can prove or show that the following happens ...
2
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1answer
56 views

Show a function is Lebesgue integrable

Hi I am struggling with a question but really I am struggling more with the concepts behind it so any help would be appreciated. Q/ Let $f(x)=x^{-\frac{1}{2}}$ for $x\in(0,1)$ and 0 otherwise. Let ...
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1answer
18 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
1
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1answer
34 views

Proving a function is Lebesgue integrable

I need to prove that $$\frac{|x|^\alpha}{1+x^2}$$ is Lebesgue integrable for $\alpha \in [0,1)$ but I'm not sure how to do this. I first tried expanding this using the Taylor expansion to show it is ...
3
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0answers
50 views

Real Analysis versus Measure & Integration

I am looking at next semester's class schedule at my school, especially at a graduate course named Measure & Integration. Officially it is described as "... an introduction to the principles, ...