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.
1
vote
2answers
161 views
Measurable Functions
How do we prove that a function $f$ is measurable if and only if $\arctan(f)$ is measurable?
If I use the definition of measurable functions, that is, a function is measurable if and only if its ...
1
vote
1answer
65 views
integrals and characteristic functions
$f$ is Lebesgue integrable over $A$, and $B$ is a measurable subset of $A$. I want to show $$\int_B f=\int_Af\chi_B$$, where $\chi_B$ is the characteristic function of $B$ (it is 1 on B and 0 ...
2
votes
1answer
277 views
Lebesgue Convergence using The General Lebesgue Dominated Convergence Theorem
Let ${f_n}$ be a sequence of integrable functions on E for which $f_n \to f$ a.e. on E and f is integrable over E. Show that $\int_E |f-f_n| \to 0$ if and only if $\lim_{ n\to\infty} \int_E |f_n| = ...
3
votes
2answers
180 views
The General Lebesgue Integral
For a measurable function, $f$, on $[1, \infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1} f$ for each natural number $n$. Is it true that $f$ is integrable over $[1, \infty)$ if ...
3
votes
4answers
139 views
Measure and Lebesgue Integral
I got this exercise as homework and I found some problems in solving it. So I hope that someone can help me.
Let $f:[0,1] \rightarrow R$ Lebesgue measurable and $S=\{x \in [0,1]:f(x) \in Z\}$.
Show ...
3
votes
1answer
321 views
{$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$
Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$.
I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$.
My attempts:
I've found ...
3
votes
1answer
61 views
Uniform integrablity of measurable functions
How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}?
$($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$
...
1
vote
1answer
94 views
Limit of a measurable function and the Lebesgue integral
Suppose $\{f_n\}$ is a sequence of lebesgue measurable functions such that $f_n\rightarrow f$, except on a set of measure $0$, as $n\rightarrow\infty$, and $|f_n(x)|\leq g(x)$, where $g$ is ...
2
votes
2answers
99 views
Trying to show that a function is zero almost everywhere given a constraint on its Lebesgue measure.
We have that $g$ is a measurable and bounded function on $[a,b]$. I have $\int_a^cg=0$ for every $c\in[a,b]$. I want to show $g=0$ on $[a,b]$ except possibly on a subset of measure zero.
Proof.
By ...
4
votes
2answers
94 views
Limit and Lebesgue integral in a compact
I have problem with the exercise that follows.
Let $(z_m)_m \in R^n$ so that $\Vert z_m \Vert \rightarrow \infty$ when $m\to \infty$.
Let $f:R^n \rightarrow [-\infty;+\infty]$ integrable.
Show ...
1
vote
1answer
319 views
Finding Lebesgue Integral of $\frac{1}{\sqrt{x}}$ over $(0,1]$
How do I rigorously discover what
$$
\int_{(0,1]} \frac{1}{x^{1/2}} = \underset{0 \le \phi \le \frac{1}{\sqrt{x}}}{\sup} \int_{(0,1]} \phi
$$
(for $\phi$ a simple function) is? Note that I have ...
1
vote
0answers
70 views
$\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm$
Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that
$$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$
Proposition: $f$ is ...
2
votes
2answers
582 views
Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$
I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
4
votes
1answer
165 views
Extension of Fatou's lemma
let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$.
I want to show if (1) holds, then (2) holds too.
$$\lim_{n \rightarrow ...
0
votes
1answer
158 views
Lebesgue Integral on a set of measure zero
I need to show that if $f$ is an integrable function on $X$ and $\mu(E)=0 ,\ E\subset X$; then $\int _E f(x) d\mu(x)=0$ .
In my attempts I've showed that $\forall \epsilon > 0 \ \ \exists ...
2
votes
1answer
96 views
using sup of an unbounded function
Is what I'm doing valid if we don't have any information on boundedness of $f$ or $f_n$?
let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions, $f_n ...
0
votes
1answer
108 views
Uniform integrability and Lebesgue convergence
A). Given that $ |X_n| \leq Y $ and $Y \in L$. Try to show $X_n$ is lebesgue integrable.
b). Try to give any example for which $X_n \longrightarrow^{L} X$ yet $\not\exists Y \in L$ with $|X_n| \leq ...
4
votes
1answer
246 views
Integration by parts and Lebesgue-Stieltjes integrals
I want to use Integration by parts for general Lebesgue-Stieltjes integrals.
The following theorem can be found in the literature:
Theorem: If $F$ and $G$ are right-continuous and non-decreasing ...
5
votes
3answers
165 views
Non-Lebesgue Integrability of $1/|x|$ over $[1, \infty)$
How does one show that $\int_\mathbb{[1, \infty)}1/|x|$ is not (Lebesgue) integrable?
What I could think of is as follows:
Letting $f(x)=1/|x|$ (defined for $|x|\geq 1$), define $f_n(x)=f\chi_{[1, ...
3
votes
1answer
68 views
What can we tell about a sequence of measurable functions on a finite measure space such that $\sup_n \int_X |f_n(x)|^2 d\mu < \infty$?
I found this on a qualifier exam, and I think it will help me understand $L^p$ spaces better.
Let $f_n$ be a sequence of measurable function on a finite measure space. Suppose that
$$\sup_n \int_X ...
1
vote
1answer
108 views
to show a function is Lebesgue integral
I need to show that $f=\frac{1}{\sqrt x}$ is Lebesgue integrable on [0,1].
My attempt:
I need to show $\sum_{m=n}^\infty \frac{m}{n} \mu(E_m^{(n)})$ converges absolutely $ \forall n$.
...
1
vote
1answer
71 views
Convergence of functions in $L^1$
I am trying to prove a theorem, and I have been able to reduce it to the following question. I feel that this should be easy, but I can't see the solution.
If $(g_n)_{n\geq 1}$ is a sequence of ...
1
vote
1answer
266 views
improper Riemann integral and Lebesgue integral
Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral $\lim_{\epsilon \to 0} ...
0
votes
1answer
84 views
What does Luzin's theorem imply?
Luzin's theorem states that:
let $f:[a,b]\rightarrow R$ be an a.e. finite function,
$f$ is measurable iff $\forall \epsilon \geq 0: \exists \phi_\epsilon$ continuous on $[a,b]$ and $\mu\{x: f(x)\neq ...
0
votes
3answers
121 views
Lebesgue- integrability of roots and powers of a function
If the powers of a function $f$ are Lebesgue integrable what can we say about the original function?
for example if we take $f=\frac{1}{x} on [1, \infty] $, it is not integrable but $f^2$ is.
Is there ...
1
vote
2answers
73 views
Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$
Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$ cX := \{ cx \mid x \in X \}. $$ Then $$ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$
Now I can prove this for ...
5
votes
2answers
600 views
Generalisation of Dominated Convergence Theorem
Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
0
votes
3answers
114 views
Is $C_0^\infty$ dense in $L^p$?
I have a question concerning the Lebesgue spaces:
Is $C_0^\infty$ dense in $L^p$ ?
And if yes, why?
Thanks!
1
vote
1answer
183 views
Easy application of the Dominated Convergence Theorem?
I am struggling with an application of the Dominated Convergence Theorem (DCT) which has cropped up a few times in various proofs I have been studying, in particular a proof about approximating ...
1
vote
2answers
67 views
Expectation and Lebesgue integration question
How might I show that if a random variable (call it Z) is such that EZ (expectation of Z) is finite (i.e. it is Lebesgue integrable), then nP(|Z|>n) tends to 0?
1
vote
2answers
352 views
Lebesgue measure sigma algebra
Lebesgue measure on sigma algebra, help ...........
Which of the following are sigma algebras? reply with justification please.
All subsets in rational numbers
{ {0},{1},{0,1} }in space {0,1}
all ...
5
votes
2answers
182 views
constructing a sequence of simple functions with Lebesgue measure approaching the riemann integral
Let $\lambda$ denote the Lebesgue measure on the Borel sets of [0,1]. Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. I know that the Riemann integral $I:=\int_{0}^{1} f(x)dx$ exists. I also know ...
1
vote
2answers
113 views
Cantor ternary set problem
Let C be a cantor ternary set
If $x,y \in C,$ then obviously $x-y \in [-1,1]$
Conversely I want to prove that
if $w \in [-1,1],$ then there exists $x,y \in C$ such that $x-y=w$
How to prove this ...
1
vote
1answer
80 views
Relation among $L^{p}(\mathbb{R}^d)$?
Let $L^{p}(\mathbb{R}^d)$ be the linear space consists of $L^p$-integrable functions on $\mathbb{R}^d$ for $1\le p \le \infty$. Are there any relation among these spaces?
1
vote
2answers
87 views
Equicontinuous, differentiable continuous problem
Assume that each of {$f_n : [0, 1] \rightarrow R$} is continuously differentiable
I know that if {$f_n'$} is uniformly bounded, {$f_n$} is equicontinuous.
However, the converse is NOT true.
I want ...
1
vote
1answer
61 views
Superposition operator in Sobolev spaces
While working on an elliptic problem in $\mathbb{R}^N$, I met an issue that I cannot work out clearly. Assume that we have a continuous function $g \colon \mathbb{R} \to \mathbb{R}$ such that
...
3
votes
1answer
112 views
Compact set in all $L_p$, $1\leq p<\infty$
Suppose $X\subseteq L_\infty$ is a compact subset of $L_p$ for all $1\leq p<\infty$. Does this mean that for every $\epsilon>0$ there exists a measurable set $E\subseteq [0,1]$ with ...
3
votes
0answers
120 views
Lebesgue Integration fundamental questions
My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
1
vote
0answers
186 views
Extended Riemann integrability of a non-negative function implies Lebesgue integrability?
Let $f$ be a bounded function on a finite interval $[a, b]$ of the real line.
If $f$ is Riemann integrable, we denote its Riemann integral by $\mathcal{R}(f , [a, b])$.
It is well known that $f$ is ...
5
votes
1answer
158 views
Yet another definition of Lebesgue integral
Let $[a, b]$ be a finite interval of the real line.
A partition $P$ of $[a, b]$ is a finite sequence of numbers of the form
$a = t_0 < t_1 <\cdots < t_{k-1} < t_k = b$
Let $(X, \mu)$ be ...
2
votes
1answer
158 views
Another definition of Lebesgue integral
Let $(X, \mu)$ be a measure space.
Let $X = A_1 \cup\cdots\cup A_k (A_i \cap A_j = \emptyset$ for $i \neq j)$, where each $A_i$ is measurable.
We say $\pi = \{A_1,\dots,A_k\}$ is a finite measurable ...
3
votes
2answers
115 views
Lebesgue generalizations of Hilbert spaces?
Is an L[p] space a generalization of Hilbert spaces using Lebesgue integration?
And if this is the case, is it true that Holder's and Minkowski's Inequalities are generalizations of the ...
1
vote
2answers
63 views
Lebesgue integration for $u \in C^{\infty}_c$
Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support.
Is $u$ in $L^2(\Bbb{R}^d)$?
I think that $u$ is in ...
3
votes
2answers
235 views
Application of Radon Nikodym Theorem on Absolutely Continuous Measures
I have the following problem:
Show $\beta \ll \eta$ if and only if for every $\epsilon > 0 $ there exists a $\delta>0$ such that $\eta(E)<\delta$ implies $\beta(E)<\epsilon$.
For the ...
11
votes
1answer
118 views
Does $f_n \to 0$ in $L^1(\mathbb R^2)$ imply that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?
I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise.
Suppose you have $(f_n)_{n\ \in \mathbb N}\subset ...
4
votes
2answers
91 views
Convergence in $L^1$ problem.
Problem: Let $f \in L^1(\mathbb{R},~\mu)$, where $\mu$ is the Lebesgue measure. For any $h \in \mathbb{R}$, define $f_h : \mathbb{R} \rightarrow \mathbb{R}$ by $f_h(x) = f(x - h)$. Prove that:
...
3
votes
1answer
192 views
A question about integral operator
I have a question: Prove or disprove that: for every $f\in L^{1}\left(\mathbb{R}\right)$, $$\sup\left\{ { ...
1
vote
2answers
290 views
Derivative of step functions
I was reading up on the Lebesgue integral and how it is computed. And since it is a generalization of the Riemann integral in a more theoretic framework, the same fundamental principle holds, only for ...
0
votes
0answers
153 views
Lebesgue Integration of Measurable Function
Can I ask a homework question here?
Let $f$ be measurable and nonnegative in $\mathbb{R}^n$
Define a radial function $f^*(|x|)=\inf\{t:\lambda(\{x:f(x)>t\})\leq|x|\}$.
Show that ...
1
vote
3answers
145 views
Lebesgue Integration problem
Can I ask a homework question here?
Assume that $f \in L^q(\mathbb R^d)$ for some $ q < \infty$ . show that
$\mathrm{lim}_{p \to \infty}||f||_p = ||f||_{\infty}$
$p$ conjugate of $q$
