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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2
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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 ...
0
votes
2answers
37 views

Prove that $f^y$ and $f_x$ are Lebesgue-integrable

Let $f:\Bbb R^2\to \Bbb R$ given by: $$f(x,y) = \begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2} & \text{if $(x,y)\in(0,1)\times(0,1)$} \\ 0 & \text{if $(x,y)\not\in(0,1)\times(0,1)$} \\ ...
2
votes
2answers
43 views

Subsequence of functions in $L^p$

On a problem sheet we were asked to find a sequence of functions $(f_n)_{n \geqslant 0} \in L^p [0,1]$ such that $\lim_{n \to \infty} ||f_n||_p = 0$ but $\lim_{n \to \infty} f_n (x)$ doesn't exist ...
6
votes
2answers
136 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 ...
9
votes
0answers
127 views

Topology of convergence in measure

Currently I am doing some measure theory(on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure) and I am looking at sets$A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
1
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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} ...
2
votes
0answers
42 views

Jump is no Lebesgue Point

Let $f$ be locally integrable, then for $x_0\in\mathbb{R}$ we have $$\lim\limits_{R\to 0}\frac{1}{|B_R(x_0)|}\int\limits_{B_R(x_0)}|f(x)-f(x_0)|dx=0.$$ The point $x_0$ is called Lebesgue point of $f$. ...
0
votes
0answers
21 views

Stieltjes integral and Lebesgue measure

Sometimes I see a Stieltjes integral where the differential is $dF(x)$, but when they derive it (assuming $F$ has a derivative) they get $F'(x)d\mu(x)$, for a Lebesgue measure $\mu$. Where does this ...
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 ...
0
votes
1answer
38 views

Lebesgue integration: Showing $\displaystyle\lim_{\lambda \rightarrow \infty} \int_{0}^{\infty} e^{-x}\cos(x)\arctan(\lambda x) \ dx = \dfrac{\pi}{4}$

I am trying to show that: $\displaystyle\lim_{\lambda \rightarrow \infty} \int_{0}^{\infty} e^{-x}\cos(x)\arctan(\lambda x) \ dx = \dfrac{\pi}{4}$ I've tried using MCT/DCT but haven't found a ...
1
vote
0answers
110 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 ...
0
votes
0answers
22 views

Can the theory of Lebesgue integration be extended in a way analogous to extending Riemann integrals to improper Riemann integrals?

I recently (last night) learned the definition of Lebesgue integration and one of the limitations I was told was that some improper Riemann integrals aren't Lebesgue integrable. It occurred to me ...
1
vote
1answer
36 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 ...
3
votes
1answer
36 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
2
votes
1answer
40 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 ...
2
votes
2answers
45 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
0
votes
1answer
39 views

There exists an open set such that $vol_n(\Omega)<vol_n(X)+\varepsilon$

Assume: $\mathscr L(\Bbb R^n)$ is the set of Lebesgue-integrable functions, $vol_n(X)=\int_{\Bbb R^n} 1_X$, and $\int^*f=\inf\{\int h: h\in S_*, 1_X\le h\}$. Let $X\subset\Bbb R^n$ and ...
0
votes
1answer
20 views

g_n integrable on R

Let g_n (x) = 1 if x=0 sin x /x if -n<= x <= n 0 if x<-n or x>n show that for every n, g_n is integrable ...
1
vote
3answers
76 views

A basic question on integration [closed]

$x^{k}{\rm e}^{-x^{2}}$ decreases to zero "exponentially" when $x \to \pm \infty$, $\int_{\mathbb R}{\rm f}\left(x\right)\,{\rm d}x < \infty$. Which theorem is being used here ?
1
vote
1answer
39 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 ...
0
votes
1answer
50 views

$f '$ is not Lebesgue integrable on $[-1,1]$

Let f be that function from R to R defined by f(x)= 0 if x=0 x^2 sin(1/x) if x not = 0 show that the function f' is ...
1
vote
1answer
36 views

A little help on properties Lebesgue integration.

Suppose $f$ is a nonnegative $\mathcal{M}-\text{measurable}$ function and $\{E\}_{n=1}^\infty\subset\mathcal{M}$ with $E_1\supset E_2 \supset \cdot \cdot \cdot $. Further suppose $\int_\mathbb{R}f \ d ...
1
vote
0answers
50 views

General Lebesgue Integral Problem

I am stumped on trying to solve the following exercise: Let f be a measurable function in E which can be expressed as $$ f = g +h$$ Where g is finite and integrable over E and h is nonnegative on ...
0
votes
1answer
23 views

Prove measurability of a function defined from two measurable functions

You have two measurable functions $L$ and $U$ defined on $([0,1],\mathcal{B}[0,1],Leb)$. Define $$ f = \begin{cases} L & \text{if }L=U \\ 0 & \text{otherwise} \end{cases} $$ The text says ...
1
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0answers
30 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
0answers
32 views

Slight issue with Lebesgue Integration (Dominated Convergence Theorem)

I have the following question: Prove that $\displaystyle\lim_{n \rightarrow \infty} \int_0^{n^2} e^{-x^2}n\sin\left(\frac{x}{n}\right) \ dx = \frac{1}{2}$ When doing this question, I showed ...
0
votes
1answer
50 views

Show that f is Lebesgue integrable on R? [closed]

Show that $f(x)=1/x^{1/2}, x \in (0,1]$ is Lebesgue integrable?
5
votes
0answers
57 views

Theorem $2.14$ page $40, 41$ in Rudin - Real and Complex Analysis

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
0
votes
1answer
59 views

Not Lebesgue Integrable

Let g(x) be the function from R to R defined by $g(x)= 1$ if $x=0$, $\frac{\sin x}x$ otherwise. Define the function $g_n (x)= g(x)$ if $-n ...
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 ...
0
votes
1answer
44 views

A question about Lebesgue measurable sets

Let $E\subset\mathbb{R}$ be a measurable set, and define $F=\{(x,y):x-y\in E\}$. Is $F$ a measurable set? why? Thanks to any one who give me a hint!
3
votes
2answers
65 views

Lebesgue integral question from wiki

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: Let $f: ...
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
2answers
85 views

Show that, $\int_A fg\ge \frac{1}{|A|}\int_A f\int_A g$

If $A\subset \mathbb{R}$ and $f,g :A\to\mathbb{R}$ are monotone increasing, Lebesgue integrable functions, then $$\int_A f(x)g(x)dx\ge \frac{1}{|A|}\int_A f(x)dx\int_A g(x)dx.$$ It looks like the ...
0
votes
1answer
42 views

Semicontinuity of this function

We get a continuous function $g:[a,b]\to\Bbb R$, and we defined: $$f(x)=\begin{cases} g(x) & \text{ if $x\in[a,b]$} \\0 & \text{ if $x\not\in [a,b]$} \end{cases}$$ Prove that $f$ is ...
1
vote
1answer
30 views

Reference for studying polar coordinate

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78. I find it somewhat terse (Maybe it's just me).. I guess this kind of transform is possible even when ...
2
votes
1answer
65 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: ...
0
votes
1answer
37 views

Every Cauchy sequence in $C([0,1])$ in the $L^2$ norm is also Cauchy in the $L^1$ norm.

I am asked to show the following: Proposition. There is a unique injection $j: L^2([0, 1]) \hookrightarrow L^1([0,1])$ which continuously extends $Id: C([0, 1]) \to C([0, 1])$. Here $L^1([0, ...
3
votes
0answers
77 views

Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
1
vote
0answers
54 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): ...
3
votes
1answer
44 views

How to demonstrate that continuous functions on a closed interval are Lebesgue integrable?

So, I'm trying to prove that if $f$ is a continuous function defined on a closed interval $[a,b]$ then it is Lebesgue integrable. Here is my attempt based on what I've found on the internet so far: ...
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
1answer
27 views

Integrate function with image $\mathbb{R}^n$

I know that for any measure space $(\Omega,\Sigma,\mu)$ and any $\Sigma$-borel-measurble function $f\colon \Omega \to \mathbb{R}$ the integral $$\int_\Omega |f(x)| \, d\mu(x)$$ is well definied. I ...
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 : ...
5
votes
4answers
450 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...