Tagged Questions

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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1
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1answer
15 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: ...
0
votes
0answers
18 views

Integral inequality norm

I have read these one. It should be simple, but i can't find any answer: Let $D\subset \mathbb{R}^N$ measureable and bounded, and let $G:D\rightarrow\mathbb{C}^{N\times K}$ be measureable. Then, the ...
3
votes
2answers
42 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 ...
0
votes
1answer
49 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 ...
2
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1answer
36 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 ...
0
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1answer
25 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 $$ ...
1
vote
1answer
35 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? ...
3
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2answers
37 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 ...
0
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2answers
12 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 ...
0
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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 ...
0
votes
1answer
24 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 ...
1
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1answer
64 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 ...
6
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1answer
180 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
2answers
34 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 ...
1
vote
1answer
46 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 $ ...
0
votes
0answers
18 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 ...
0
votes
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
votes
1answer
29 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 ...
1
vote
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 ...
2
votes
1answer
27 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)| = ...
0
votes
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 ...
0
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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 ...
0
votes
1answer
19 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 ...
1
vote
3answers
48 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
votes
1answer
23 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 ...
0
votes
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} ...
0
votes
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
votes
1answer
54 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 ...
1
vote
1answer
16 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
vote
1answer
31 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
votes
0answers
48 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, ...
1
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0answers
34 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
0
votes
1answer
32 views

Orthogonality of Hermite functions

I would like to prove to myself that Hermite functions, defined by $\varphi_n(x)=(-1)^n e^{x^2/2}\frac{d^n e^{-x^2}}{dx^n}$, $n\in\mathbb{N}$ are an orthogonal system in $L^2(\mathbb{R})$, i.e. that, ...
2
votes
1answer
44 views

Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$

Let $f \in L^1(\mathbb{R}^d)$. Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$. What I want to do is bound $|f(x)-f(x-t)|$ above by something and then use the Lebesgue Dominated ...
0
votes
1answer
23 views

Almost everywhere (surely) properties

In Lebesgue integration, why is it so important to have properties usually true almost everywhere ? Is it because a function like $1_{\mathbb{Q}}$ is not integrable with Riemann integration ? I am not ...
0
votes
0answers
28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
1
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0answers
40 views

Sentences about Lebesgue integrable function

In $\mathbb{R}$ with Lebesgue measure, we take $f\in L^1$ and we set $\hat{f}(t)=\int f(x) e^{ixt} dx$, for each $x$ $\ \ \ (i^2=-1)$ Show that: $\hat{f}$ is continuous $\lim_{t\rightarrow \pm ...
1
vote
1answer
63 views

Convergence as for the norm [duplicate]

If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $\|f_n\|_p \rightarrow \|f\|_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to ...
1
vote
1answer
16 views

Restriction of a finite measure to a set on unbounded function

So I have a measure $(X,\mathscr{F},\mu)$, possibly finite, or $\sigma$-finite, or a completely general finite measure. $B\in \mathscr{F}$ is a set of finite measure. For every measure set $A\in ...
0
votes
0answers
18 views

$f$ square-summable on $X'\times X''$, $\varphi_m$ square-summable on $X'$ and $\int f\cdot\bar{\varphi}d\mu'$ square-summable on $X''$

Let $X:=X'\times X''$ be the product of measure spaces $(X',\mu')$ and $(X',\mu'')$, endowed with the Lebesge extension $\mu:=\mu'\otimes\mu''$ of product measure $\mu'\times \mu''$ defined by ...
0
votes
0answers
23 views

Is it necessary to take the inferior limit of this sequence of integrals

Suppose $A $ is a measurable set, and $\{h _n \} $ is a sequence of nonnegative simple functions such that $h _n \uparrow \chi _A $, where $\chi _A $ is the charachteristic function. I wonder why in ...
0
votes
1answer
13 views

Can I change the order of summation here?

Is it true that $\sum _{i = 1 } ^n \alpha _i \sum _{r=1 } ^{\infty } \mu(A _i \cap E _r ) = \sum _{r = 1 } ^{\infty } \sum _{i=1 } ^{n } \alpha _i\mu(A _i \cap E _r ) $ I know that I can change ...
4
votes
2answers
82 views

What is the correct definition of Area?

How is the area of a rectangle: length $\times$ breadth? We know that other areas can be derived from it. Also, the area under curves uses the area of rectangles as a basis.
0
votes
1answer
17 views

composition of measureable function with $\sqrt x$ in $L^2[0,1]$

I was wondering if my hypothesis is correct: Let $X=L^2[0,1]$, $f\in X$, $g:[0,1]\to[0,1]$ definited as $g(x)=\sqrt x$ Is $f\circ g\in L^2[0,1]$ necessarily? Thanks a lot
0
votes
1answer
60 views

Composition of measureable function with continuou function in $L^2[0,1]$

I was wondering, for general knowledge, if this claim is correct. Let $X=L^2[0,1]$, $f\in X$, $g:[0,1]\to[0,1]$ invertible. Particularly, it's image is $[0,1]$ so everything is well defined. ...
0
votes
1answer
30 views

A bounded function is Riemann integrable over [a,b] and its Rieman integral equals its Lebesgue integral

In special, M0 in Lemma 6.26 denotes the family of all step functions on real line. Dear friends, I wonder whether Sj's are needed in the proof of Lemma 6.26. Personally, I believe that S0 (S ...
1
vote
2answers
57 views

Lebesgue Integral: Convexity

Given a probability measure $\rho(\Omega)=1$. Consider a complex function $f\in\mathcal{L}(\rho)$. From the Riemann integral it is evident that: $$\int_\Omega f\mathrm{d}\rho\in\overline{\langle ...