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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5
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1answer
99 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
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0answers
32 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
27 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
75 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
28 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
48 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?
3
votes
1answer
55 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
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0answers
20 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
51 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
43 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
17 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
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1answer
25 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
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1answer
34 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
179 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
29 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 $f_n$ ...
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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
23 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
42 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
20 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
49 views

Complex Measures: Pushforward

Problem Given Borel spaces $X$ and $Y$. Consider a complex measure: $$\mu:\mathcal{B}(X)\to\mathbb{C}$$ Regard a pushforward: $$h\in\mathcal{B}(X,Y):\quad\nu:=\mu\circ h^{-1}$$ Then one has: ...
3
votes
1answer
32 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
38 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
58 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
24 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
31 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
59 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
31 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
54 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
41 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
21 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
57 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
52 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
59 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
32 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?
2
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1answer
30 views

monotonic linear functional on $C_+(X)$

Let $X$ be a compact metric space. Let $C_{+}(X)$ be the set of all continuous non negative functions on $X$. Let $\lambda : C_{+}(X) \to [0,\infty)$ such that ...
0
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1answer
24 views

Does this contravene the dominated convergence theorem?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. $f(x):=\lim\limits_{n \rightarrow ...
0
votes
1answer
30 views

How to show $\int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu$

$\{f_n\}$ are nonnegative monotonic increasing functions. Show that $$ \int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu $$ Can someone give me a hint on how to show this? ...
3
votes
1answer
81 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Show that $\forall ...
1
vote
0answers
24 views

How can I show that the function in my problem is lebesgue integrable?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. Now: How can I show that ...
2
votes
1answer
34 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h ...
0
votes
1answer
31 views

weak topology and weak* topology on $L^1, L^{\infty}$

Suppose $L^1(I)$ is the primal space and $L^{\infty}(I)$ is the dual. Could I simultaneously define weak topology on $L^1(I)$ with respect to $L^{\infty}(I)$ and define weak or weak* topology on ...
0
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1answer
33 views

Prove that for almost every $x\in$ $\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$ .

Let $f\in$$L^1(\mathbb{R})$ and let $p>0$ . Prove that for almost every $x\in$$\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$.
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0answers
31 views

Completion of C(I) to $L^{2}(I)$ for some arbitrary interval I

As $L^{2}(I)$ is the completion of C(I), without too many issues (as $L^{2}(I)$ is a space of equivalence classes with equivalence relation defined as functions equivalent if differ at only finitely ...
3
votes
1answer
33 views

Monotone convergence theorem in a special case

Suppose $C^{+}[0,1]$ be the set of all continuous functions with domain $[0,1]$ taking non-negative values only. Let $\lambda : C^{+}[0,1] \to [0,\infty)$ be a map that satisfies ...
2
votes
1answer
25 views

Equivalences of weak convergence in $\mathcal{L}_p$ spaces with the Lebesgue measure

Let $\Omega =(0,1)$, and $f,f_n\in \mathcal{L}_p(\lambda)$. Prove that if $\sup_n{\| f_n \|}<\infty$ and $$\int_{(0,t]}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{(0,t]}f \, ...
2
votes
1answer
46 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
0
votes
1answer
33 views

The dual space of a product space

Suppose $\prod^n L^1(I) $ is the product space of n $L^1$ integrable functions. What would be its dual space of continous linear functionals? would it be $\prod^n L^{\infty}(I)$? Do I need the norm ...