1
vote
0answers
23 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
1
vote
0answers
33 views

Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
1
vote
1answer
34 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
1
vote
0answers
46 views

What means: is equivalent to?

I found the following theorem: Let $(f_n)$ be a sequence of norm one functions in $L^p, p \in [1, \infty)$. If $\lambda(supp(f_n)) \rightarrow 0$, then some subsequence of $(f_n)$ is equivalent to a ...
0
votes
1answer
33 views

Lebesgue integration: Existence of double integral, but not Lebesgue integrable.

I am trying to determine whether or not $f(x,y) = \dfrac{\sin(x)\sin(y)}{x^2+y^2}$ is integrable on $E = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ ...
1
vote
1answer
42 views

Lebesgue Integration: Double Integral (Fubini)

I'm trying to determine whether or not $f$ is integrable on $E$, where $f(x,y) = e^{-xy}$ and $E = \{(x,y) : 0 < x < y < x+x^2\}$ Ok, so $f$ is continuous and non-negative on $E$ so it is ...
2
votes
1answer
57 views

Want to show that a function is integrable

So here is my question, I would like to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R_+}\frac{sin(x)}{x}(e^{-x/n}-1)dx$$ To interchange the integral an the limit I want to ...
1
vote
2answers
51 views

I have a limit whose variable is both within the integral and in the integral boundaries. May I split it?

More concretely, I have the integral $$\lim_{n\to\infty}\int_{\left(0,\frac{n}{2}\right)}x^2e^x\left(1-\frac{2x}{n}\right)^nd\lambda(x)$$ It is clear that this is the same as ...
6
votes
2answers
132 views

Evaluating $\displaystyle\int_{0}^{\pi/2}\log\left(a^2\cos^2\left(x\right)+b^2\sin^2\left(x\right)\right)$

I am trying to evaluate the integral below by differentiating through the integral. Let $ F(a,b) :=\displaystyle\int_{0}^{\pi/2}\log\left(a^2\cos^2\left(x\right)+b^2\sin^2\left(x\right)\right)$ For ...
1
vote
0answers
45 views

Integration question measure theory

For the function $$ f(x) = \begin{cases} \infty & \text{if $x=0$} \\ 1/x & \text{if $x \in \mathbb{Q} \smallsetminus 0$} \\ 0 & \text{Otherwise} ...
1
vote
2answers
43 views

Dominated convergence under weaker hypothesis

Let $f_n,\,n\in\mathbb{N}$ be a sequence of real integrable functions, $f_n\to f$ pointwise as $n\to\infty$. The dominated convergence theorem states that if there exists $g\in L^1$ such that ...
1
vote
0answers
109 views

Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
1
vote
1answer
35 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
2
votes
1answer
39 views

Fundamental Theorem of Calculus for Riemann and Lebesgue

Quick question regarding the second part of the Fundamental Theroem of Calculus in terms of Riemann and Lebesgue Integration: In terms of applying the second part of fundamental theorem of calculus, ...
1
vote
0answers
83 views

Differentiation through the integral sign (Lebesgue integration)

I have to evaluate $$\int_0^{\frac{\pi}{2}}\log(a^2\cos^2x+b^2\sin^2x)dx.$$ Now I have arrived at the answer by separating the original integral into integral $\log(a^2\cos^2x)$ plus integral of ...
1
vote
1answer
38 views

Using Riemann integral to define Lebesgue Integral

In the text I'm working through, the Lebesgue integral is related to the Riemann integral as follows: For some non-negative, real valued function $f$ on $\Bbb{R}$, set $E_y=\{x:f(x)>y\}$ and ...
1
vote
0answers
29 views

Inverse map measurable

We said that a function $f:X \rightarrow \mathbb{R}$ is measurable iff we have that for all $I_a:=(a,\infty)$, $a \in \mathbb{R}$ $f^{-1}(a,\infty)$ is measurable. Now I want to show that ...
0
votes
1answer
25 views

Convergence of Integrands and Integrals

Suppose $E \subset \mathbb{R}$ is compact. Is it possible to find a sequence of positive continuous functions $f_n: E \to \mathbb{R}$ such that for every $x \in E$ we have $$f_n(x) \to f(x)$$ for some ...
2
votes
1answer
51 views

monotone convergence question

I am trying to show that $$\lim_{n \rightarrow \infty} \int^{n^2}_{0}{e^{-x^2}n \sin\frac{x}{n}dx} = \frac{1}{2}.$$ I have tried by using the monotone convergence theorem, but if I take $f_n = ...
2
votes
1answer
64 views

Understanding Lebesgue Integration

I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral: In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: ...
1
vote
0answers
53 views

Fat Cantor-Lebesgue function

I came across the following theorem the other day, "If $f:[a,b]\to \mathbb{R}$ is monotonic increasing, then $f$ is differentiable a.e." If the take the standard Cantor-Lebesgue function then I see ...
2
votes
2answers
83 views

If $f:[0,1]\to[0,\infty)$ is Riemann-Integrable on every closed subinterval of $(0,1]$, Is it possible that $f$ is not Lebesgue-Integrable?

If $f\colon[0,1]\to[0,\infty)$ is Riemann-Integrable on every closed subinterval of $(0,1]$, Is it possible that $f$ is not Lebesgue-Integrable ? According to a lemma(which has to be proven): ...
1
vote
1answer
41 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
0
votes
1answer
25 views

Integral Estimate Using a Function and its Inverse

I want to show the following: given a measure space $(X,\mu)$ and $f,g$ $\mu$-measurable functions on $X$, $$\int_X |f(x)g(x)| d\mu(x) \leq \frac{1}{2}\int_{|f(x)| \leq 1} |f(x)|^2 d\mu(x) + ...
0
votes
1answer
60 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
0
votes
0answers
27 views

Construction of Lebesgue integral

I have a couple of questions regarding the construction of the Lebesgue integral. I am looking at one construction based on simple functions that reads: Definition: A measurable function $f : ...
1
vote
1answer
44 views

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$ I have no idea how to start the exercise like that.
3
votes
2answers
72 views

Definition of the Lebesgue integral in terms of simple functions with finite measure support

Let $(X,\Sigma,\mu)$ be a measure space. A function of the form $$ \phi(x) = \sum_{i=1}^{n} c_i \mathbf{1}_{E_i} $$ where $c_i \in \mathbb{R}$ and $E_i \in \Sigma$ is called a simple function. If ...
1
vote
4answers
63 views

Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]

Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]. *I'm thinking about this but without progress...
5
votes
0answers
80 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
5
votes
3answers
359 views

Prove that function is not Lebesgue integrable

Prove that function $f(x,y)=\dfrac{1}{x^2+y^2}$ is not Lebesgue integrable on $A=(0,1]\times(0,1]$. To my knowledge the fastest way to do it is to use Fubini's theorem. From what I would get: ...
0
votes
1answer
23 views

Transformation formula for multidimensional integral

Let $A,B$ be positive definite symmetric $n\times n$ matrices. I stumbled upon the following identity and don't see why it should hold: $$\int_{\mathbb{R}^n}\frac{1}{\sqrt{\det A \det ...
0
votes
3answers
60 views

$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$ for $f \in H(\Omega)$

Let $\Omega $ be a domain ,$\overline{D(z,r)} \subset \Omega $, $f$ holomorphic in $\Omega$. a) Show that $$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$$ where $dm$ ...
0
votes
1answer
46 views

antiderivative of lebesgue integrable function

Let $f:[0,b] \to \mathbb{R}$ be Lebesgue integrable. We define $$g(x)=\int_x^b\frac{f(t)}{t}dt,\quad 0<x\le b.$$ Show that $g(x)$ is Lebesgue integrable in $[0,b]$. Also show that ...
11
votes
3answers
149 views

$\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$ for every positive definite matrix $A$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}e^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}\!, $$ where ...
7
votes
0answers
242 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
3
votes
2answers
57 views

Shift Operator not continuous

Let $X=L^1(\mathbb{R}^n)$ and $T:\mathbb{R} \rightarrow L(X)$, such that $(T(\tau)f)(t):=f(t+\tau)$. The question is: Is $T$ continuous? Well, my idea was the following: $|\tau_1-\tau_2| \le \delta ...
3
votes
0answers
35 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
1
vote
2answers
67 views

Finding a limit of an integral

I am trying to find the following limit. Let $X = [0,\infty)$ and $\mathbb B$ denote the Borel subsets in $[0,\infty)$, $\lambda$ the Lebesgue measure. Let $f_n : [0, \infty) \to \mathbb R$ be given ...
1
vote
1answer
46 views

Is each Lebesgue integrable function Riemann integrable almost equal?

There are numerous functions which are Lebesgue but not Riemann integrable, the most famous one probably being $$ f: [0,1] \rightarrow \mathbb{R}, \quad x \mapsto \begin{cases} 1, & x \in ...
0
votes
1answer
30 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
1
vote
0answers
49 views

Bounded $L^p$ functions

It is well known that we do not have $L^q(\mathbb{R}^n) \subset L^p(\mathbb{R}^n)$ for $q >p$. But is this relation true, if we assume that we only look at bounded functions $f$. I think it could ...
2
votes
2answers
71 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x ...
5
votes
2answers
172 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

I'm asked to prove $$\displaystyle\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}\tag{$\ast$}$$ by integration of $e^{-x}\text{sin}(2xy)$ over an suitable measurable ...
2
votes
1answer
50 views

Invariance of integral

Given the Lebesgue integral with the Lebesgue measure and the Borel-Sigma Algebra, I am supposed to figure out under which transformations $\int_{\mathbb{R}^2} f(x) dx$ the integral is ...
2
votes
1answer
74 views

Integration, Lebesgue and counting measure

Could you help me with the following exercise? Consider $X=Y=[0,1]$ with Lebesgue measure $m$ on $X$ and counting measure $\omega$ on $Y$. Let $f:X \times Y \rightarrow \mathbb{R}$ and $f(x,y)= ...
1
vote
1answer
58 views

Theory of “integration” on sets of measure zero (the measure zero sets are with respect to Lebesgue measure )

Suppose we have Lebesgue measurable set E. Let F be a subset of E with measure zero with respect to the Lebesgue measure on E. My question is, can we construct a "reasonable" integration theory on the ...
2
votes
1answer
63 views

Question regarding Lebesgue Integrability in $\sigma$ -finite spaces

I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable ...
1
vote
1answer
62 views

Integral of the product of an bounded and any continous functions

If $u:[0,1]\rightarrow \mathbb{R}$ is bounded measurable function so that for all $v\in C[0,1]$ $$\int_0^1uvdx=0$$then show that u is zero almost everywhere on $[0,1].$ Thanks in advance for any ...
1
vote
1answer
51 views

Give a example in Lebesgue integral

Please help me to solve the following problem that is in the Lebesgue integral discussion Give an example of a sequence $f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\|f_n\|_\infty ...