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.

learn more… | top users | synonyms

1
vote
1answer
34 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...
0
votes
1answer
16 views

condition for which we have an integrable function

Let $\Omega=[-L,L] \subset \mathbb{R}$, and let $n=\dfrac{u_x}{|u_x|}$. Now my question is what are the conditions on $\gamma(n)$ and $u_x$ so that we have $$\gamma^2(n) u_x \in L^1$$ i.e. ...
2
votes
2answers
65 views

Lebesgue integrability and measurable functions

Let $f$ be a nonnegative function on the reals. What does the (Lebesgue) measurability of $f$ have to do with the (Lebesgue) integrability of $\int f$? I've spent some time studying the definition at ...
2
votes
0answers
31 views

Proving this is Lebesgue integrable using radial functions

Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue ...
0
votes
0answers
21 views

How can I calculate the following Lebesgue integral $\int_{-\infty}^{\infty}||x|^{-0.35}-|x-1|^{-0.35}|^{1.8} d\lambda(x)$

How can I calculate the following Lebesgue integral? Is it convergent? $\int_{-\infty}^{\infty}||x|^{-0.35}-|x-1|^{-0.35}|^{1.8} d\lambda(x)$ where $\lambda$ is the Lebesgue measure.
0
votes
1answer
61 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
39 views

If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
1
vote
1answer
77 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 ...
0
votes
1answer
25 views

lebesgue integral

let $f\ge 0$ be a measurable function s.t. $\int_R fdm=\infty$, show that for any M>0 there is a real measurable function g, and $0\le g \le f$ and the following hold: $\int_R g dm \ge M$ and g is ...
0
votes
1answer
42 views

lebesgue measure and integral

assume that f is a non-negative real-velaued measurable function, and $\int_R f(x)dm<\infty$ (lebesgue measure) show for any real number a, a is not 0, $\int_R f(ax)dm=\frac{1}{|a|} \int_R f(x)dm$
3
votes
1answer
41 views

Lebesgue integral over “bad” measurable sets

Let $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) be a bounded open domain and $f \in L^\infty(\Omega)$ possibly changes the sign. Assume that the set $$ \Omega^+ := \{x \in \Omega: f(x) > 0 \} $$ has ...
2
votes
1answer
31 views

Understanding the proof of completeness of $L^1$.

I'm reading the proof of completeness of $L^1 (X, \mathscr{M}, \mu)$, and I would like to clear up some confusion To prove $L^1$ is complete it suffices to show that every Cauchy sequence $(f_n)$ has ...
0
votes
1answer
61 views

Lebesgue integrable function without compact support.

Suppose $f \in L^1(\mathbb{R}, m)$, where $m$ is Lebesgue measure. By definition we have $$\int_{\mathbb{R}} f dm < \infty$$ Does $f$ have compact support? This makes sense, but I don't know if ...
0
votes
1answer
39 views

Integrability of sums of Dirac deltas

this is my first post in the forum and I am an engineer, so I apologize in advance if my question is not clearly stated. Consider the function $f(x)=\sum_{i=1}^N a_i\delta(x-x_i)$ where ...
1
vote
1answer
46 views

No Tonelli&Fubini contradiction

I was trying to solve the following question: Let $f(x,y)=\cases{1/x^2: x>y\ge0\\ -1/y^2: y>x\ge0\\ 0: x=y}$ Show that $\intop_0^1dx\intop^1_0f(x,y)dy\ne\intop_0^1dy\intop_0^1f(x,y)dx$ I did ...
-1
votes
2answers
39 views

Lebesgue integral over Infinite measure sets

I would apreciate if someone could tell me wether this is true or false, or any advice on how to prove it or disprove it: Let $f$ be a positive measurable function over $(X,S)$ where S is a ...
3
votes
1answer
55 views

Integration by parts in Bochner Lebesgue spaces.

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, ...
2
votes
1answer
64 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
58 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
43 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
53 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
171 views

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

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)\,{\rm ...
12
votes
2answers
225 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
vote
0answers
52 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
46 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
28 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
49 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
62 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
1answer
122 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
33 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
45 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
38 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
60 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
90 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
48 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
49 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
21 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
49 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
62 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
46 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
61 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
26 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
vote
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
50 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
58 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
73 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
63 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
27 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
64 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!