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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-3
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1answer
23 views

How can I prove that $f$ and $g$ are measurable functions [on hold]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
2
votes
2answers
60 views

Prove that $\iint\limits_ {[0,1] \times [0,1]} \frac{x^2-y^2}{(x^2 + y^2)^2}\,\mathrm dx\,\mathrm dy$ is not integrable

I have to prove that the following integral does not exist: $$\iint \limits _{[0,1] \times [0,1]} \frac{x^2-y^2}{(x^2 + y^2)^2}\,\mathrm dx\,\mathrm dy .$$ I think I can use Fubini's Theorem, ie. if ...
5
votes
2answers
46 views

If one side of $\int f\ d\lambda = \int f\ d\mu - \int f\ d\nu$ exists, does the other one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
0
votes
1answer
37 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
1
vote
0answers
18 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
2
votes
1answer
18 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
4
votes
1answer
55 views

$L^{\infty}$ norm

For Lebesgue $p$-integrable functions, what would be the formula for $$\left(\int_0^1 \sum_{i=1}^n | f_i(x)|^p dx\right)^{\frac{1}{p}} $$ as $p\to +\infty$? Would it be $$\max_i \sup_{[0,1]} ...
0
votes
1answer
51 views

Lebesgue differentiation

Some days back I was doing the lebsegue integration. I was amazed by the integration's ability to maximize the potential of Reinman integration. Are there any new differentiation (except the metric ...
1
vote
1answer
29 views

Lebesgue integral of a ratio of Lebesgue densities

I need a hint to solve the following problem: $P$ is a probability mass on $\mathcal B(\mathbb R)$ with a Lebesgue density $h$, $f$ is another Lebesgue density. I need to show that $\int ...
0
votes
0answers
19 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
0
votes
0answers
13 views

Proof of sets to integrate over in Green's, Stokes and Divergence Theorems

As the title says I've some doubts about these theorems, because my definitions of them are so abstract, I'm struggling to apply them. For Green's theorem, in order to be able to apply it, the region ...
5
votes
1answer
90 views

Lebesgue Integration Question

Let $f$ be integrable with respect to a Lebesgue measure. Evaluate the limit, $$\lim_{n \to \infty} \int_{-\infty}^{\infty} f(x-n)\left(\frac{1}{1+|x|}\right)\,dx$$ I tried change of variables ...
0
votes
0answers
30 views

Relation between $\lim_{a \to 0}\int_a^T u(t)$ and the Lebesgue integral $\int_0^T u(t)$

Let $u\colon (0,T] \to \mathbb{R}$ be function with $u \geq 0$ everywhere and $u$ is continuous on $[a,T]$ for every $a > 0$. Suppose that the limit $$\lim_{a \to 0}\int_a^T u(t) \;dt ...
0
votes
0answers
21 views

Proving convergence in L1 of a sequence of functions given by integrals

I am required to prove that $x\mapsto\int_x^{x+1/n} n f(y)dy$ converges in the $L^1$ sense to $f$, knowing that $f\in L^1$. My current attempt is: after a variable change, I've rewritten ...
7
votes
2answers
63 views

Lebesgue integrable function over $(0,1)$ vs $[0,1]$

Up till now, I thought saying $u \in L^2([0,1])$ is the same as saying $u \in L^2((0,1))$, because I see people emphasizing "$u$ is Lebesgue integrable over $[0,1)$". I thought the whole point of the ...
0
votes
0answers
25 views

Approximation Lemma for Riemann-integrable functions

In the following let $f : [a,b] \to \mathbb R$ be bounded functions. For a regulated function, the integral could be written as the limit $$ \int_a^b f(x) dx = \lim_{n\to \infty} \int_a^b ...
0
votes
2answers
39 views

Bounded function on compact interval that is not Lebesgue integrable

Is there an example of a bounded function $f : [a,b] \to \mathbb R$ which is not Lebesgue integrable?
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votes
0answers
19 views

What are equivalent way of defining Lebesgue integration? [closed]

Is there any book or lecture notes which tells all the equivalent way of defining Lebesgue integration?
3
votes
1answer
51 views

Compute the following Lebesgue Integral

I've been sitting on this question for a while now: Let $f(x) \in L^1(\mathbb{R})$. Compute $$ \lim_{h\to \infty} \int_{\mathbb{R}} |f(x+h)-f(x)|dx. $$ I've managed to convince myself that the ...
0
votes
0answers
18 views

A problem about a family of mesurable fuctions

Let $(X,\mathcal{M},\mu)$ a measure set such that $\mu(X)<\infty$ and $\mathcal{F}$ a family of $\mu$-measurable functions. Let $E(f,t)=\{x\in X\mid f(x)\geq t\}$ with $f\in\mathcal{F}$. If ...
1
vote
1answer
48 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
1
vote
1answer
37 views

Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.

Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $. Problem. Show that $ f(x) = 1 $ ...
0
votes
1answer
13 views

Definition of lebesgue integral with respect to measure $\mu$ [duplicate]

In Rudin's Real and Complex Analysis, the Lebesgue integral is defined as: L et $(X,m,\mu)$ be a measure space, where $X$ is a set, $m$ is a $\sigma$ algebra on $X$ and $\mu$ is a measure. Then, if ...
1
vote
1answer
22 views

Partial Integration for measures

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular ...
-1
votes
1answer
28 views

Borel Measures: Coproduct

I need this thread as lemma! (See the advice: SE: Q&A) Given Borel spaces $\Omega_\lambda$. Consider the coproduct: ...
2
votes
3answers
125 views

A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$. However Royden & Fitzpatrick’s book "Real Analysis" (4th ...
1
vote
1answer
28 views

A sequence tending to zero where each element cannot be bounded by an integrable function

I am teaching myself measure theory and I working through http://homepages.uconn.edu/~rib02005/real.html. In exercise 7.3, they ask: Give an example of a sequence of non-negative functions $f_n$ ...
-1
votes
0answers
95 views

Example: $|f|$ is integrable but $f$ is not integrable

Can someone give me an example about 1. A function $f$ that is not measurable but $|f|$ is measurable 2. A function $f$ that is not Lebesgue integrable but $|f|$ is Lebesgue integrable. (This ...
1
vote
0answers
32 views

Collection is uniformly integrable, but individual is not integrable

Could you give me an example about: "a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable." This sounds counterintuitive? However ...
1
vote
1answer
21 views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Problem Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon>0$. Prove that $\{f_n\}$ is uniformly integrable. Background A family $\{f_n\}$ ...
2
votes
1answer
37 views

Lebesgue integration

if $f : \mathbb{R} \to \mathbb{R}$ is continuous function which is Lebesgue integrable on $\mathbb{R}$ then show that there is sequence $x_n$ which goes to infinity and $x_n f(x_n)$ goes to $0$. ...
1
vote
1answer
17 views

$\|\sum_ia_i g(x-i)\|_{L^p(\mathbb{R})}\le (\sup_x \{\sum_k|g(x-k)|\})\|a\|_{L^p(\mathbb{Z})}$

Let $a=\{a_i\}$ be an arbitrary sequence of complex numbers with finitely many non-zero terms. Consider the function $f(x)=\sum_ia_i g(x-i)$, where $g$ is a good function. Prove that for any $p\in ...
2
votes
1answer
37 views

Complex Measures: Pushforward

Attention I added a hypothesis! Given measure spaces $\Omega$ and $\Pi$. Consider a complex measure: $$\mu:\Sigma(\Omega)\to\mathbb{C}:\quad\mu\left(\biguplus_kA_k\right)=\sum_k\mu(A_k)$$ Regard a ...
3
votes
1answer
29 views

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

Background Let $E_M=\{x: |f_n(x)>M\}$. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon >0$, there exists $M$ such that $$\int_{E_M} |f_n| \ ...
2
votes
1answer
30 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
3
votes
1answer
56 views

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $u\in C^0(\Omega)$ and $(u_n)_{n\in\mathbb{N}}\subseteq C_0^0(\Omega)$ with $$\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0\tag{1}$$ Can ...
0
votes
1answer
20 views

Bounded Variation imply L^1

If a function $f:\mathbb{R}\to\mathbb{C}$ is of bounded variation, is it true that $f\in L^1(\mathbb{R})$? We need to ask for $f$ to be continuous?
2
votes
1answer
27 views

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Problem Statement Let $\{f_n\}$ be a sequence of real-valued, measurable functions on $[0,1]$ that is uniformly bounded. Show that if $A$ is a Borel subset of $[0,1]$ then there exists subsequence ...
0
votes
1answer
29 views

Evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$

I have to evaluate $ \int_{\mathbb{R}^n} \! \exp(-||x||^2) d\mu$ as part of another problem. Can someone give me a hint on how to do this?
2
votes
1answer
34 views

Prove that $\sum_{k=0}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \mathrm{ d}x$

Problem Statement Prove that $$\sum_{k=1}^{\infty}\frac{1}{(k+p)^2}=-\int_0^1\frac{x^p}{1-x}\log x \ \mathrm{d}x$$ Background I just learned the limit theorems (MCT, LDCT, Fatou's Lemma). This ...
2
votes
0answers
31 views

Show $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$

$A$ is a invertible $n \times n$ matrix. Show: $\int_{\mathbb{R}^n} \exp(-\|Ax\|^2) d\mu(x)=\frac{\pi^{\frac{n}{2}}}{|\det(A)|}$ Can someone give me a hint on how to show that?
3
votes
3answers
54 views

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Background I seem to be having issues recognizing valid bounding functions when applying the Lebesgue Dominated convergence theorem. Here are two examples I did that I do not think are justified. ...
0
votes
3answers
26 views

Equivalent way of writing the norm of Lp

Given a measurable $E\subset \Bbb R^d $ and a measurable function $f:E\rightarrow \Bbb R^d $, prove that : $$ \int (\left\lvert f \right\rvert)^r d\mu = r\int_{0}^\infty t^{r-1} \mu(\{x \in E ...
2
votes
1answer
42 views

Failure of Newton-Leibniz formula

Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable but $f \notin C^1 ( \mathbb{R} )$ . It means that $f'$ exist but it is not continuous. Question 1 Is function $f'$ locally ...
3
votes
1answer
40 views

Is $\lim_{p \searrow 1} \|u\|_{L^p(\Omega)} = \|u\|_{L^1(\Omega)}$ true?

Let $\Omega \subset \mathbb{R}^n$ be a bounded, open set and $u \in L^q(\Omega)$ for some $q \gt 1$ (ie. $u \in L^p(\Omega)$ for all $p \in [1,q]$). Is then $$ \lim_{q \ge p \searrow 1} \left( ...
1
vote
1answer
20 views

Monotone Convergence theorem for monotone decreasing sequences

Short question: (Just an example. I want to know if similar thoughts can be used for other sequences of functions) If I want to evaluate $\lim_{n\rightarrow \infty}\int_{[0,1]}-nxdx$, I can't do that ...
0
votes
1answer
53 views

Complex Measures: Polynomials

Given the complex plane $\mathbb{C}$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\operatorname{supp}\mu\subseteq\overline{B_r}$$ Then one has: ...
0
votes
1answer
46 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
4
votes
1answer
57 views

What is so good about the $L^2$-norm of the second derivative being small?

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' ...
0
votes
0answers
31 views

Relation between $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ Norms of a Lebesgue function

Can $\mathcal{L}_{1}$ norm of a function $f(t)$ be related with its $\mathcal{L}_{2}$ norm as $||f||_{1} \leq ||f||_{\infty} ||f||_{2}$ or something like this?