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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40 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: ...
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0answers
43 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
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58 views

$\lim\limits_{n\to\infty}\displaystyle\int_X n\log((1+(f/n)^{\alpha})d\mu$

suppose $\mu$ is a positive measure on $X$ and $f:X\to[0,\infty]$ is measurable with $\int_Xfd\mu=c$, where $0<c<\infty$ and let $\alpha$ be a constant, prove that; ...
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1answer
56 views

Show that $\frac{1}{x^4 \sin^2 (x) +1} \in L^1([0, \infty))$

This is question 10.20c from Apostol's Mathematical Analysis. Basically, I am trying to show that $$ f(x)=\frac{1}{x^4 \sin^2(x)+1} \in L^1([0,\infty)) $$ I know that for some value $k$, I can ...
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1answer
39 views

Integral convergence

Please how can I show that the $\lim_{n \rightarrow \infty}\int_{\Re^+}f_n d\mu$ converges and determine its limit in the following cases of $f_n: \Re^+ \rightarrow \Re$ (a)$f_n(x) = sin(nx) ...
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2answers
72 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
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1answer
126 views

Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
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1answer
22 views

Show the $\min(f_n, f)$ has lebesgue integral converge to $\int_{A}f dm$

let $A$ be a subset of the reals, $f_n,f$ are positive lebesgue measurable functions that $f_n$ converge to f pointwise and $\int_{A}f_n dm$ converge to $\int_{A}f dm$ with $\int_{A}f dm<\infty$. ...
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1answer
48 views

Lebesgue Integrabilityon relative measure space

This shouldn't be too difficult but I just can't get started. There are quite a few similar problems on my problem set so I am hoping to get some idea here on this one, and try solving the others ...
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1answer
35 views

Measure Theory - Convergence of functions with bounded integrals

A question I came across. Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Let $f_1,f_2,\dotsc:X\to\mathbb R$ be measurable functions such that $n^2\cdot\lVert ...
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0answers
30 views

Measure Theory - Lebesgue Integral over non- $\sigma$-finite spaces

In most courses on Measure Theory the Lebesgue Integral is introduced initially for simple functions on finite spaces, then for general functions on finite spaces and finally for general functions on ...
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45 views

Measure Theory - An identity for the Lebesgue Intgral

I'm trying to solve the following exercise in Measure Theory: Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Prove that for every $0\leq f\in L^{1}(\mu)$ it holds that: ...
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1answer
32 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
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1answer
30 views

prove that the lebesgue integral of 2 functions is finite

For the first function we have: if $\mu(X)\lt\infty$ and $f\in L^+$ then $\int fd\mu\lt\infty \iff \sum_{n=0}^\infty 2^n\mu(\{x\in X:f(x)\ge2^n\})\lt\infty$ For the second one: let $f\in L^+$ ...
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1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
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1answer
35 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
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1answer
53 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
54 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
31 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
64 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
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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
60 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 ...
0
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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 ...
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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, $\int_R f(ax)dm=\frac{1}{|a|} \int_R f(x)dm$
3
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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
45 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 ...
3
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1answer
53 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 $(x,y)\not\in(0,1)\times(0,1)$} \\ ...
2
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2answers
51 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
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2answers
170 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
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
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0answers
51 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
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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 $f$. ...