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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1answer
55 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 ...
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0answers
35 views

lebesgue integral of $f(x^n)$

I know that $f:[0,1]\to \mathbb{R}$ is continuous at $0$, and $f\in L_1([0,1])$. How can one prove that $f(x^n)\in L_1([0,1])$, for any $n\in \mathbb{N}$?
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1answer
55 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
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2answers
36 views

Lebesgue integral of a non-negative function.

I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301. In this problem we are given $f$ a nonnegative and integrable function on ...
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1answer
24 views

For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$?

The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values ...
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1answer
50 views

f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
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2answers
131 views

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
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0answers
48 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 ...
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1answer
32 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists ...
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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 ...
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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. ...
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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 ...
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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 ...
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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.
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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 $E = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ ...
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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.
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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 ...
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1answer
26 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 ...
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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, $\int_R f(ax)dm=\frac{1}{|a|} \int_R f(x)dm$
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
48 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 ...
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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 ...
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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
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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 ...
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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 ...
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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 $(x,y)\not\in(0,1)\times(0,1)$} \\ ...
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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 ...
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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
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2answers
226 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 ...
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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} ...
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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 $f$. ...
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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 ...
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2answers
50 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 ...
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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 ...
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1answer
123 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 ...
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0answers
34 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 ...
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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 ...
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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 ...
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1answer
61 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, ...
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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 ...
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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),\ ...
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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
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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 ...
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3answers
77 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
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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
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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 ...
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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 ...