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
votes
3answers
100 views

Inherent Pitfall of Lebesgue Integration?

I am studying Real Analysis with Royden's Book. I noticed that for a function f differentiable almost everywhere on [a, b] and f' integrable over [a, b], it does not imply that $ f(x) = \int_{[a, ...
2
votes
1answer
52 views

a question about the evaluation of integral [duplicate]

Let $\alpha:[0,1] \to R$ be the Cantor function. Evaluate $$\int_{0}^{1}xd\alpha $$and $$\int_{0}^{1}x^2d\alpha.$$ I know that the Cantor function is continuous and monotone increasing, how can I ...
1
vote
0answers
55 views

Puzzles in a proof

From a previous link in MSE: Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11), the question states Suppose that $\{n_k\}$ is an increasing sequence ...
1
vote
1answer
50 views

Existance of limit and Integrability of a function

The questions is: Let $f$ be a function defined as $f(x) = (-1)^n/n $ for $x \in [n, n+1), n \in \mathbb{N}.$ Show that $lim_{n\to\infty}$ $\int_{[1,n]}\ f $ exists Also, is $f$ integrable on ...
0
votes
0answers
40 views

Show that Thomae Function is Lebesgue Integrable

I have the Thomas Function defined as follows: $f(x): [0,1] \to \mathbb R$ $f(x) = q$ if $x$ is rational and $x = p/q$, $0$ otherwise (please note that this is the usual definition of THomae's ...
0
votes
0answers
23 views

Integrability of a function, $(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$.

Is the function $$(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$$ integrable wrt. the lebesgue measure on $(\mathbb{R}^{2},\mathbb{B}_{2})$? I have shown that it's not integrable, ...
6
votes
2answers
97 views

Question about Dominated Convergence Theorem.

How to compute $$\lim_{n \to \infty}\int_0^{\infty}\Big(1+\frac{x}{n}\Big)^{-n}\sin\Big(\frac{x}{n}\Big) dx$$ I want to use the Dominated Convergence Theorem. so it becomes ...
2
votes
2answers
65 views

integral over a subset of $\mathbb {R}^2$ is not defined while…

consider the function $f(x,y)=\frac{xy}{(x^2+y^2)^2}$, we can see by some easy calculation that $\int_{-1}^1\int_{-1}^1 f(x,y)\,dx\, dy$ and $\int_{-1}^1\int_{-1}^1 f(x,y)\,dy\, dx$ exist and equals ...
2
votes
1answer
49 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
1
vote
0answers
26 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
0
votes
1answer
42 views

prove the equivalence between a null set and a limit

I'm asked to prove that for any non-negative, measurable and integrable function $f$ on $[0,1]$, we have $\lim\limits_{a\to 0}\int_{0}^{a}fdx=0$. I want to use the theorem that for null set E, such a ...
2
votes
1answer
55 views

Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.

Let $g$ be a non-negative measurable function. For $1 \leq p < \infty$, show that $$\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$$ where $\mu$ is the Lebesgue measure and we are ...
2
votes
2answers
41 views

Comparing limits of integrals

If $$f_n:X\rightarrow [0,\infty]$$ is a sequence of measurable functions and we know that $$\lim_{n\rightarrow \infty }\int_X f_n \,d\mu=0,\qquad \qquad \tag{$\star$}$$ then can we conclude that ...
4
votes
1answer
85 views

Limit of the integral $\int_0^1\frac{n\cos x}{1+x^2n^{3/2}}\,dx$

Prove that $\displaystyle\int_0^1\frac{n\cos x}{1+x^2n^{\frac32}}dx\rightarrow0$ as $n\rightarrow\infty$. $f_n(x)=\frac{n\cos x}{1+x^2n\sqrt{n}}$ tends to zero function pointwise. It just ...
6
votes
1answer
40 views

how to prove $\int{f}d\mu=\sum_{x\in\Omega}f(x)$

Prove that $\int{f}d\mu=\sum_{x\in\Omega}f(x)$ when $f$ is absolutely summable, where $\mu$ is a counting measure on the measure space $(\Omega,\mathscr{F})$. Can someone give me hints?
0
votes
1answer
35 views

Definition of Lebesgue integrable function

If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a ...
4
votes
2answers
54 views

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$. I have seen a proof of this already in a lecture but I ...
6
votes
2answers
99 views

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e.

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e. I let $E \subset \mathbb{R}^d$ be a finite measurable set. I try to break this into two cases: Case 1: If $f(x)=0$ ...
1
vote
1answer
25 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
5
votes
1answer
92 views

Trying to calculate the integral limit $\lim_{n\rightarrow\infty} \int_{-\sqrt n}^{\sqrt n}\left (1 - \frac{x^2}{2n}\right)^ndx$

How to calculate following integral: $$\lim_{n\rightarrow\infty}\int_{-\sqrt{n}}^{\sqrt{n}}{\left(1-\frac{x^2}{2n}\right)^n}dx$$ Prove that this integral exists and compute its value. I just ...
1
vote
0answers
22 views

improvement of upper Lebesgue sum

In Pugh's real mathematical analysis, lower and upper Lebesgue sum are given as: $\underline{L}(f,Y)= \sum_\limits{i=1}^{\infty}y_{i-1}\cdot mX_{i-1}$ $\overline{L}(f,Y)= ...
1
vote
1answer
51 views

show Lebesgue dominated convergence theorem fails for ${n^2xe^{-nx}} x\in [0,1]$

show Lebesgue dominated convergence theorem fails for the sequence of functions $f_n=n^2xe^{-nx}$ $x\in [0,1]$ Here is my solution. Is it correct? $f_n$ is an integrable function the sequence ...
2
votes
2answers
38 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
3
votes
0answers
132 views

Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
2
votes
2answers
49 views

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?
-1
votes
2answers
32 views

Is this subspace of $L^1(\mathbb{R},m)$ closed? [closed]

Let $K$ be the subspace of $L^1(\mathbb{R},m)$ which contains precisely the functions such that $\int f=0$. Is $K$ closed? (EDIT: When I asked this question, I could only see that ${f:||f||_1=0}$ is ...
3
votes
0answers
53 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
1
vote
0answers
20 views

property of distribution function

Let $f$ a continuous map from $\mathbb{R} \rightarrow \mathbb{R}$ and let $L_1, L_2$ 2 probability measures on $\mathbb{R}$. Let $K$ be a closed set in $\mathbb{R}$. In a proof, I want to use the ...
1
vote
2answers
33 views

Application of dominated convergence theorem- find limit

Find (with justification) $$ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx $$
2
votes
0answers
38 views

What does it mean when $\sigma$ is discriminatory?

I am reviewing this paper on the approximating power of neural networks and I came across a definition that I could not quite understand. The definition reads: where $I_n$ is the $n$-dimensional ...
0
votes
1answer
11 views

Convergence in absence of DCT

Can you give an example of a sequence of non-negative functions tending to zero pointwise such their integral tends to zero but there is no integrable function which bounds them?
1
vote
0answers
32 views

Checking $g(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)\in L_1$ for $r_i\in\mathbb{Q}$

Let $x^{-1/2}$ for $0<x<1$ and $f(x)=0$ otherwise. Let $(r_n)$ be an enumeration of $\mathbb{Q}$ and let $g(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)$. Show that $g\in L_1$ and in particular, $g$ is ...
0
votes
1answer
28 views

Lebesgue Integrable functions

I am in need of guidance for the following question: Let $f:X\to\mathbb{R}$ be an integrable function. Show that $\mu(\{x:|f(x)|\geq n\})\leq 1/n\int |f|\mu(dx)$ for each $n>0$.
2
votes
3answers
62 views

Showing $\int_E f=\lim_{n\to\infty}\int_E f_n$ for all measurable $E$

The following is an exercise from Carothers' Real Analysis: Suppose $f$ and $f_n$ are nonnegative, measurable functions, that $f=\lim_{n\to\infty} f_n$ and that $\int f=\lim_{n\to\infty}\int ...
4
votes
2answers
61 views

Is there a notion of indefinite Lebesgue integral?

When I started studying integration rigorously via the Riemann and Lebesgue integrals, one thing that struck me is that we loose completely the concept of indefinite integrals. Integrals of functions ...
3
votes
1answer
79 views

Lebesgue integral and iterated integral

I am learning lebesgue integral at the moment, and come across a question in homework, but find it really confused. The question states: I first tried to compute the iterated integral by Riemann ...
2
votes
0answers
103 views

upper lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable.Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...
1
vote
0answers
30 views

Prove weak derivative commutes with difference quotient

Let $U$ be an open set in $\mathbb{R}^n$,$f:U\to \mathbb{R},f\in W^{1,p}(U)$. Let $\tau_{h,i}f(x)=\frac{f(x+he_i)-f(x)}{h},h>0$ Given any compact $V\subset U$, show there exists $h_0>0$ such ...
2
votes
1answer
46 views

Lebesgue integral of absolute value of sequence of functions [duplicate]

I am working on a problem$^{(*)}$ on Lebesgue integral looks like this: Given that both $f_n$ and $f$ are integrable, $f_n \longrightarrow f$ a.e., and $\int|f_n| \longrightarrow \int |f|$. Show ...
3
votes
1answer
65 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
6
votes
1answer
173 views

Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and strongly converge to $u$ in $L^p(\Omega).$ We define a function $f:\Omega\times ...
0
votes
2answers
180 views

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why and why isn't this true for $1/x$? I'm having difficulty understanding difference between the above functions in terms of ...
1
vote
1answer
47 views

Confused about switching Lebesgue integrals for Riemann integrals

Hi I have been attempting given in the link below. I am confused about the argument used to show the function is not Lebesgue integrable. This question What each person has used to answer is the ...
3
votes
1answer
45 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
3
votes
2answers
46 views

Proving that $m(E) = 0$ if for all $n$, $\int_E x^n \cos x\, d x = 0$

Suppose that $E\subset [0,2\pi]$ is measurable and $\int_E x^n \cos x\,dx = 0$ for all $n =0,1,2,\cdots$. Then prove that $m(E)=0$. In a non-rigorous fashion, if $\sum_{1}^{\infty} a_nx^n = \sec ...
0
votes
3answers
117 views

Is a compact set an union of a finite number of disjoint closed intervals?

I think it is true for $\mathbb R$ with usual metric. How about others? How to prove it? Motivation: I got this idea when I was reading a proof for Lebesgue's Criterion of Riemann Integrability, here ...
2
votes
0answers
44 views

If a simple function is measurable, then is its characteristic function

I am working on a problem looks like this: If a simple function $s$ is measurable, show that its characteristic function $\mathcal X_{X_i}$ is measurable. Here are the ways I have been working ...
1
vote
2answers
107 views

Lebesgue Integral: $\int_1^{\infty}\frac{1}{x}$

The following is an exercise from Carothers' Real Analysis: Show that $$\int_{1}^{\infty}\frac{1}{x}=\infty$$ (as a Lebesgue Integral). Attempt: Let $E=[1,\infty)$. $\int_E f=\int f\cdot ...
0
votes
0answers
36 views

A question about countably subadditive property of Lebesgue Outer Measure

Here is the definition of Carothers' Lebesgue Outer Measure: . And countably subadditive property of Lebesgue Outer Measure has been talked here: I can understand all proofs. However, I'm ...
0
votes
1answer
38 views

How will m*(rE) behave?

Let $rE =\{rx: x\in E\}$, what is $m^*(rE)$ in terms of $m^*(E)$? Intuitively, I think $m^*(rE)\leq r\times m^*(E)$. However I've no idea how to prove it? Add: Definition of Lebesgue Outer Measure ...