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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4
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1answer
48 views

A Measure Theory problem-If $\int_{A_n}f(x)dx\rightarrow0 $ then $\lambda(A_n)\rightarrow0$

This question was proposed as part of a test for PhD applicants but considered too hard and rejected. I tried unsucessfully to solve it for quite some time. For anyone wishing to try his luck.. ...
1
vote
1answer
45 views

Is $\iint \dfrac{1}{z} dxdy\neq 0$?

I am trying to solve an exercise and at some point I came accross the integral $$\iint_L \dfrac{1}{z} dxdy,$$ ($z=x+iy$) where $L\subset \mathbb{C}$ is a compact set with positive two-dimensional ...
1
vote
1answer
41 views

Is it true that $\lim\limits_{r \rightarrow 1}f(r x) = f(x)$ in $L^1$?

Suppose $f \in L^1(\mathbb{R})$ with Lebesgue measure and $r > 0$. Does $f(rx)$ converges to $f$ in $L^1$ as $r \rightarrow 1$ ? Put differently, does $$ \| f(rx) - f(x)\|_1 \rightarrow 0$$ as $r \...
0
votes
1answer
25 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
votes
1answer
44 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 $t\in J \subseteq \mathbb{R}$. 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 ...
5
votes
1answer
69 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
vote
2answers
23 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
27 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 $\int_E f$...
0
votes
1answer
69 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
vote
0answers
43 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 $$ I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz $$ ...
2
votes
1answer
36 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 ...
0
votes
1answer
43 views

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

Notation) $\mathbf{x} = (x_1, \cdots, x_n)$, $\mathbf{y} = (y_1, \cdots, y_m)$, $I_1=\{\mathbf{x}: a_i\le x_i\le b_i, ~~i=1, \cdots, n\}$ $I_2=\{\mathbf{y}: c_j\le y_j\le d_j, ~~j=1, \cdots, m\}$, ...
1
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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
vote
1answer
27 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
votes
3answers
44 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
28 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
99 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 $(...
4
votes
0answers
27 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 ...
1
vote
1answer
44 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
45 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 $\displaystyle\...
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 ...
0
votes
0answers
37 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 ...
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 \...
2
votes
0answers
27 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
votes
1answer
48 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
votes
0answers
23 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
votes
1answer
44 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
votes
1answer
38 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 $\...
1
vote
1answer
24 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 $\mu(G|H)<\epsilon$.
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
21 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 ...
0
votes
1answer
22 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 $$\int_{\...
1
vote
0answers
56 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 $\mu_\Bbb N(A)= \vert {A \cap \Bbb N}\vert$...
0
votes
1answer
59 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 $\mu_\Bbb N(A)= \vert {A \cap \Bbb N}\vert$...
4
votes
1answer
71 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 $\...
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 $\...
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 ...
1
vote
1answer
69 views

Limit of a sequence of Lebesgue integrals

Let $ f\in L^{1}(E) $ and $ \{E_n\}$ be a sequence of measurable subsets of $E$. If $$ \lim_{n\to +\infty} m(E_n) = 0$$ prove that $$ \lim_{n\to +\infty} \int_{E_n} f = 0.$$ I tried to interchange ...
3
votes
1answer
27 views

Lebesgue integration and uniform convergence

Let $\Omega$ be a bounded and measurable set in $\mathbb{R}$. If $\{f_n\}$ is a sequence of bounded and Lebesgue integrable functions on $\Omega$. If $f_n$ uniformly converges to $f$, then how to ...
0
votes
1answer
35 views

Using the monotone convergence theorem to show a function is integrable

Apply the monotone convergence theorem and the fundamental theorem of calculus to show that $f(x) = \left\{ \begin{array}{ll} x^{-a} & \mbox{if } 0 < x \leq 1 \\ \infty & \mbox{if } ...
4
votes
0answers
84 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, ...
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 ...
1
vote
1answer
25 views

Non-regular measure can be represented by a regular measure

Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular). Then the linear map $L : C_c(X) \to \Bbb C$ ...
3
votes
1answer
13 views

Product of weak and strong convergent sequences in $L^p$

I already saw some proofs here with $b_n\to b$ in $L^2$ and $a_n\rightharpoonup a$ in $L^2$. Then $$ \int a_n b_n \to \int a b. $$ But what goes wrong if both sequences are weak convergent? Proof: $...
1
vote
0answers
24 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 ...
0
votes
0answers
24 views

Integrals as Signed Measures (and vice Versa)?

1. Can every integral (with respect to an integrable function) be written as a signed measure? And does the function’s decomposition into positive and negative parts align somehow with the ...
2
votes
1answer
174 views

Principal value integral of complex exponential

I'm reading the article Brownian distance covariance and stumbled upon a equality I can't seem to derive myself. We are first presented with the following lemma: and after stating this lemma, the ...
0
votes
0answers
19 views

Measurability and integrability of set and function

My textbook said: Let $E\subset\mathbb{R}^n$, let $G$ be an open set, and let $|\cdot|_e$ denote outer measure. if $\exists{}G$ s.t. $E\subset{}G$ and $|G-E|_e\lt\varepsilon$ for an any given $\...
0
votes
1answer
29 views

$f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. Find $\lim \int f_n d\lambda$.

Let $X = \mathbb R$, $\textbf{X} = \textbf{B}$ and $\lambda$ the Lebesgue measure on $\textbf{X}$. I have the following: $f_n = (\frac{1}{n})\chi_{[n, +\infty)}$. I need to find the following: $\...
1
vote
3answers
46 views

$f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$

If $f(x)$ and $xf(x)\in L^2(\mathbb{R})$ then $f(x)\in L^1(\mathbb{R})$. I know that if $E$ is of finite measure, then we can infer from $f(x)\in L^2(E)$ to get $f(x)\in L^1(E)$. However, now $E=\...