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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0
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1answer
16 views

Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
0
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0answers
10 views

Bounding Problem /Conditions for the Lebsgue Integral of a Function depending on two parameters to be continuous

let $F(t)=\int_{E} f_t(x)$ for $t\in J \subseteq R$. Then some theorem says that $F$ is continuous if $1)$ $\forall t_0$ $f_t(x) \rightarrow f_{t_0}(x)$ as $t \rightarrow t_0$ almost everywhere on ...
1
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0answers
16 views

Suppose that there is a finite $c$ such that $\int_0^1 | f(a+t) - f(b+t) | dt \le c$ for all $a$ and $b$. Show that $f \in L(0, 1)$.

Please help me understand the following proof. Q) Let $f$ be measurable and periodic with period $1$, that is, $f(t+1)=f(t)$. Suppose that there is a finite $c$ such that $$\int_0^1 | f(a+t) - ...
4
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0answers
47 views

Prove that $f\in L^2$ and $\lim_{n\rightarrow\infty} \int_A f_n = \int_Af$

Let $A$ be a bounded, measurable susbset of $\mathbb{R}$. Prove that if $(f_n) \subset L^2 (A)$ converges uniformly to $f$ on $A$, then $f\in L^2(A)$ and $\lim_{n\rightarrow\infty} \int_A f_n = ...
1
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2answers
18 views

For what $q$ is $\frac{\sin(x)}{x^q}$ Lebesgue Integrable on (0,1] where $q>0$

You can show $\frac{1}{x^q}$ converges on$ (0,1] $ for $q<1$ and that's a bound for the $\frac{|\sin(x)|}{x^q}$ so we know for $q<1$ our function is integrable- I can't seem to improve on this ...
1
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0answers
22 views

Is an integrable function always measurable?

There is a theorem in my textbook. Theorem $5.1$ Let $f$ be a nonnegative function defined on a measurable set $E$. Then $\int_E f$ exists if and only if $f$ is measurable. Notation If $\int_E ...
0
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1answer
65 views

what does $\frac{\text{d}x}{x}$ mean?

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure ...
1
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0answers
41 views

Evaluating a limit of an integral

I have a function $f(x,y,z) :\mathbb{R}^3 \rightarrow \mathbb{C}$, a smooth function. I know that $$ I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz $$ ...
1
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1answer
32 views

Using MCT twice to show the limit of an integral depending on $x$ and $n$

So I have $\displaystyle\lim_{n \to \infty} \int^{n^2}_0 e^{-x^2} n \sin\left(\frac{x}{n}\right) dx$. I'd like to apply the MCT but the trouble is there is a limit which also depends on $n$ So I ...
5
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1answer
56 views

Converse for Fubini-Tonelli's theorem

By Fubini-Tonelli's theorem, we know that if $E\in \mathbb{R^{n+m}}$ and $f: \mathbb{R^{n+m}}\to \mathbb{R_{>0}}$ are measurable and $f$ integrable, then the sections $E_x=\{y\in \mathbb{R^m}: ...
0
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1answer
35 views

What is the property $\mathfrak{F}$ in Fubini's theorem?

Notation) $\mathbf{x} = (x_1, \cdots, x_n)$, $\mathbf{y} = (y_1, \cdots, y_m)$, $I_1=\{\mathbf{x}: a_i\le x_i\le b_i, ~~i=1, \cdots, n\}$ $I_2=\{\mathbf{y}: c_j\le y_j\le d_j, ~~j=1, \cdots, m\}$, ...
2
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2answers
681 views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and ...
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0answers
25 views

Contradiction to a Theorem for Lebesgue integrability proof

We know that $f$ is Lebesgue Integrable iff $|f|$ is Lebesgue Integrable. We have shown by contour integration that $\int^{\infty}_0 \frac{sin(x)}{x}$ is $\frac{\pi}{2}$ - yet we can also show using a ...
1
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1answer
25 views

Examples of functions where the Lebesgue integral as a measure is complete.

Let $f\in\mathcal{M}(\mathbb{R})$ non negative. For each $E\subset\mathbb{R}$ measurable we define $\mu_f(E)=\int_{E}f$. Prove (a) $\mu_f$ is a measure in $\mathcal{M}$ (b) Give an example of a ...
3
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3answers
43 views

Why is $\frac{1}{x}$ not Lebesgue integrable on $[0,1]$?

My teacher said (without explaining) that $\frac{1}{x}$ is not Lebesgue integrable on $[0,1]$? Could someone please explain why is this true?
3
votes
1answer
26 views

Lebesque integral is not injective?

I know that Lebesgue integral is not injective i.e. if $$ \int\limits_{\Omega} f\, d\mu=\int\limits_{\Omega} g \,d\mu $$ then it is not necessary that $f=g$ on $\Omega$, But is there a simple ...
3
votes
2answers
97 views

Possible to do better than an upper bound for$\int^{\infty}_0 e^{-x}\log(x)\ dx$?

I used the series expansion of $e^{-x}$ and the fact that $\log(x)$ was less than $x$ in the $(0, \infty)$ to get an upper bound and so use simple comparison to show this was indeed integrable over ...
0
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1answer
36 views

Lebesgue integral, path connected and compact

Let $K \subseteq \mathbb R^d$ be path-connected and compact and $f:K\to\mathbb R$ continuous. How can I show that there is a $\xi\in K$ such that $$\int_Kfd\lambda^d=f(\xi)\lambda^d(K)$$ where ...
4
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0answers
24 views

If $P$ and $Q$ are Lebesgue partitions then is $P\cap Q$ a refinement to $P$ and $Q$?

My teacher said that if $P$ and $Q$ are Lebesgue partitions then $P\cap Q$ is a refinement to $P$ and $Q$. But shouldn't $$P\cup Q$$ be the refinement to both $P$ and $Q$? Or, are both $P\cup Q$ and ...
2
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1answer
40 views

Please check whether the proof is correct or not.

Please check my solving. I want to know where to be wrong or illogical, or where logical jumps are. Problem Let $y=Tx$ be a nonsingular linear transformation of $\mathbb{R}^n$. If ...
1
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1answer
42 views

Question about Lebesgue integral

When I prove a real analysis problem, I need a theorem about Lebesgue integral, but I cannot find this theorem in any standard reference, intuitively, I think it is correct, but I do not know how to ...
2
votes
1answer
163 views

Suppose that all functions ${f_n},f$ are integrable. Is $\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f.$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
0
votes
1answer
60 views

if $\int{}f$ is finite, then $\int{}f$ exists?

My textbook said, If $\int_E f$ exists then, of course, $-\infty\le\int_E f\le+\infty$. If $\int_E f$ exists and is finite, we say that $f$ is Lebesgue integrable, or simple integrable, on $E$ and ...
1
vote
1answer
29 views

Is $\mathscr{C}(K)$ a subspace of every $L^p(K)$, if $K$ is compact?

Considering a function $f\in\mathscr({C}(K),\lvert\lvert \cdot\rvert\rvert_\infty)$, i.e.$$f:K\rightarrow\mathbb{C}$$ everywhere continuous where $K$ is a compact subset of $\mathbb{R}^n$, does $f$ ...
49
votes
2answers
16k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
0
votes
1answer
18 views

When summation of two sequences is finite, is one finite?

$|\cdot|$ is Lebesgue measure. Let $w(\alpha) := |\{x:f(x)>\alpha\}|$ Let $f$ be a nonnegative function. Then, the proof uses that $\displaystyle\sum_{k=-\infty}^{\infty} 2^{kp}w(2^k)\lt\infty ...
0
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0answers
36 views

$f'$ Lebesgue-integrable

Let $f:[a,b]\to\mathbb R$ be differentiable and the derivative $f'$ bounded. How to show that $f'$ is Lebesgue-integrable on $[a,b]$ and $$\int_{[a,b]}f'd\mu=f(b)-f(a)$$ where $\mu$ denotes the ...
1
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0answers
22 views

How is justified the derivation under the integral sign $\frac{d}{d\sigma} \left( \Re\frac{1}{\zeta(s)} \right) $?

Taking $\sigma=\Re s>1$ (this is we take $s=\sigma+it$, $\sigma$ and $t$ real numbers) then the using theknown integral representation for $\frac{1}{\zeta(s)}$, where $\zeta(s)$ is the Riemann ...
2
votes
0answers
26 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
0
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1answer
30 views

Three questions on measurable functions and $L^p$ spaces

I'm learning about measure theory and $L^P$ spaces and need help with the following questions: True or False (justify): $(1)$ Let $f:(-1, 1) \to \mathbb{R}$ measurable on $(-n, n), \; \forall ...
0
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0answers
21 views

Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
0
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1answer
41 views

Can someone solve my non-understandable process in proving a theorem?

Theorem. Let $E$ be a subset of $\mathbb{R}^n$. Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$. For your ...
0
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1answer
52 views

Exist measurable functions $f_n$ with $\limsup_{n\to\infty} f_n (x) = \infty$ but $\lim_{n\to\infty} \int f_n = 0$?

Are there measurable functions $f_n: [0,1] \to [0, \infty)$, $n \in \mathbb{N}$, with $$\limsup_{n\to\infty} f_n (x) = \infty \qquad \forall x \in [0,1],$$ but $$\lim_{n\to\infty} \int_{[0,1]} f_n (x) ...
0
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0answers
16 views

Interesting measure theory property in L^p [duplicate]

Let $f, f_n \in L^p (X)$, so that there is a function $g\in L^p (X)$ with $|f_n|\leq g,\ \forall n$ and $\forall \epsilon>0, \lim_{n\to\infty} \mu (\{x\in X\big | |f_n (x)-f(x)|\geq \epsilon\})=0$. ...
1
vote
1answer
386 views

limits of an integrable function over increasing sequence of sets

Let $(E_n)_{n \geq 1}$ be an increasing sequence of sets such that $\bigcup_{n \geq 1} E_n = \Omega$. Then for every integrable function $f$ we have $$\lim_{n \rightarrow \infty} \int_{E_n} f d\mu = ...
1
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1answer
18 views

$A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$

Let A be a real set then is it true that $A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$.
1
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1answer
22 views

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
0
votes
1answer
18 views

Continuity of Integration (Lebesgue)

On the theorem regarding continuity of integration: Let $f$ be integrable over $E$. If $\{E_{n}\}^{\infty}_{n=1}$ is an ascending countable collection of measurable subsets on $E$, then ...
0
votes
1answer
35 views

How to evaluate the Lebesgue integral of the Heaviside function?

I have to evaluate the Lebesgue integral $$ I = \int\limits_{[-1, 1]} \chi(x) \chi(x - \frac{1}{2}) d\left(\chi(x)\chi(x + \frac{1}{2})\right) $$ where $ \chi $ is the Heaviside function: $$ ...
2
votes
1answer
18 views

If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable?

If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable? I know that this is false unless $f_{n}$ converges uniformly to $f$, but is there an example ...
3
votes
1answer
280 views

Lebesgue integration and countable partitions

The book "A Primer of Lebesgue Integration" by H.S. Bear defines Lebesgue integration through lower and upper sums $L(f,P) = \sum m_i\mu(E_i)$ and $U(f,P)=\sum M_i\mu(E_i)$ where infinite countable ...
0
votes
1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
0
votes
4answers
70 views

$\int_\Omega f d\mu = 0 $ if and only if $f(x)=0$ almost everywhere

can someone give me a hint on what kind of theorem/definition I should make use of to solve this? Let $(\Omega,\mathfrak A, \mu)$ be a measure space and $f:\Omega \to \mathbb R$ a non-negative ...
1
vote
0answers
55 views

Comparison test and DCT

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
5
votes
1answer
64 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
4
votes
0answers
81 views

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
1
vote
1answer
392 views

Holder inequality (reverse or equality?)

For bounded $\Omega\in\mathbb{R}^n$, it is easy to see by the Holder inequality that $\int_{\Omega} u\,dx\leq (\int_{\Omega} 1^2\,dx)^{\frac{1}{2}} (\int_{\Omega} ...
1
vote
1answer
58 views

$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$

How can I prove $$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$$ given a measure space $(\Omega,\mathfrak A, \mu)$, a non-decreasing sequence $(f_n)$ of measurable functions on ...
16
votes
3answers
1k views

How to decide whether Lebesgue integral or Riemann integral?

Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebegue integral? For instance, the Gamma ...
4
votes
1answer
36 views

Definition of outer Measure

As I understand it, the outer measure $\mu^{*}(A)$ is used to find the length of the smallest cover that covers $A$. However, in another definition, the outer measure is defined as the largest lower ...