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.
11
votes
2answers
2k 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 ...
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 ...
10
votes
5answers
353 views
Is there a fundamental reason that $\int_b^a = -\int_a^b$
Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
8
votes
6answers
136 views
give an example of a function that is integrable in $\mathbb R $ and $\lim_{ x\to \infty}f(x)\neq0$
i did a search for such function but didn't found anything useful/complete ! , like this :
Integrable function $f$ on $\mathbb R$ does not imply that limit $f(x)$ is zero
is there any function that ...
8
votes
3answers
256 views
How do I prove $f=0$ almost everywhere?
During one of the problems in Rudin I was asked to show $f=0$ a.e. Here $f$ satisfies this condition:
$$f(x)=\frac{1}{x}\int^{x}_{0}f(t)dt$$ almost everywhere and is in $L^{p}(0,\infty)$. So constant ...
8
votes
1answer
89 views
Holder's inequality for infinite products
In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of ...
8
votes
1answer
440 views
I want to understand uniform integrability in terms of Lebesgue integration
According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ ...
7
votes
1answer
92 views
$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n ...
7
votes
2answers
148 views
Measure theory questions
i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim ...
7
votes
3answers
179 views
Convergence of functions in $L^p$
Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
7
votes
0answers
178 views
Difference of differentiation under integral sign between Lebesgue and Riemann
Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign.
Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
6
votes
3answers
456 views
Can a function that has uncountable many points of discontinuity be integrable?
First of all, I would like to show you how we defined Riemann-integrals and Lebesgue-integrals to make sure that we are talking about the same:
Riemann-intregrability
Let $f:\mathbb{R} \rightarrow ...
6
votes
1answer
144 views
How to find a measurable but not integrable function or a positive integrable function?
For an arbitrary interval $I$, how can we find a positive on $I$ integrable function? And how does one construct a measurable but not integrable function.
If not all measurable functions are ...
5
votes
3answers
112 views
$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$
If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would ...
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? ...
5
votes
1answer
233 views
Uniform Integrability
Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$.
I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that
$$ \displaystyle ...
5
votes
1answer
93 views
Equi-integrability of a single function: is it the same as summability?
Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties:
For all $\varepsilon > 0$ there ...
5
votes
2answers
111 views
Derivative of $t \mapsto \Vert f+tg \Vert_p^p$
Suppose $(X,\mathcal A, \mu)$ is a measure space and let $f,g\in L^p(X)$ be real-valued functions, $p\in(1,+\infty)$. Let us define
$$
F:\mathbb{R} \ni t \mapsto \int_X \vert f(x)+tg(x) \vert^p ...
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 ...
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, ...
5
votes
0answers
51 views
Various integration theories
Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely:
Riemann Integration
Riemann-Stieltjes Integration
...
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 ...
4
votes
1answer
426 views
Limit of $L^p$ norm
Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
4
votes
1answer
234 views
Integral vanishes on all intervals implies the function is a.e. zero
I am having trouble with the following problem:
$f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$:
$$\int_{[0,a]}f\,dm=0.$$
Prove that $f=0$ for $m$ almost every $x$ ...
4
votes
2answers
155 views
How to prove that $\frac{\sin x}{x}$ is not Lebesgue integrable on $[0,+\infty]$?
How to prove that $\displaystyle\int_0^{+\infty}\left|\dfrac{\sin x}{x}\right| \, dx = +\infty$ ?
Could any one give some hint ? Thanks.
4
votes
2answers
157 views
Lebesgue integral and sums
I'm not particularly well read on the Lebesgue integral, but I have heard that it permits a much wider class of functions, and in particular we can interchange integrals and limits more easily. ...
4
votes
2answers
60 views
Question from Folland on modes of convergence
I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated.
Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow ...
4
votes
1answer
253 views
How to use lebesgue dominated convergence theorem to find the limit of an integral
I have to use lebesgue dominated convergence theorem to prove that
$$
\lim_{n\rightarrow\infty}\int_0^\infty \left[1+ \frac{\ln(x + n^2)}{n^{1/2}}\sin(x^2) + \cos\left(\frac{1}{n+x}\right)\right] ...
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:
...
4
votes
1answer
85 views
Homogenous measure on the positive real halfline
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
4
votes
1answer
304 views
Bounded convergence theorem
Suppose that $f_n$ is a sequence of measurable functions that are all bounded by M, supported on a set E of finite measure, and $f_n(x)\to f(x)$ a.e. x as $n\to \infty$. Then f is measurable, bounded, ...
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 ...
4
votes
2answers
87 views
Infinite shots fired in a lattice forest
A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
4
votes
1answer
35 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
4
votes
1answer
161 views
Sum of Lebesgue integral (absolutely integrable functions)
This is Tao Analysis II Prop. 19.3.3. (b)
Let $\Omega \subseteq \mathbb R^n$ measurable and $f,g: \Omega \rightarrow \mathbb R$ absolutely integrable functions. Then $f+g$ is absolutely integrable ...
4
votes
2answers
71 views
On an identity about integrals
Suppose you have two finite Borel measures $\mu$ and $\nu$ on $(0,\infty)$. I would like to show that there exists a finite Borel measure $\omega$ such that
$$\int_0^{\infty} f(z) d\omega(z) = ...
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.
$$\lim_{n \rightarrow ...
4
votes
1answer
248 views
Integration by parts and Lebesgue-Stieltjes integrals
I want to use Integration by parts for general Lebesgue-Stieltjes integrals.
The following theorem can be found in the literature:
Theorem: If $F$ and $G$ are right-continuous and non-decreasing ...
4
votes
0answers
98 views
Three properties of the Lebesgue measure on $\mathbb{R}^n$
I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$.
It is a non-negative ...
4
votes
0answers
484 views
Dunford-Pettis Theorem
The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that:
A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact.
Now ...
3
votes
2answers
120 views
Suppose that for all $g\in L^2(\mathbb{R}^d)$ we have $\int f_{n}g \to \int fg $. Prove convergence in $L^2$ problem.
How to solve the following problem:
Let $f_n, f \in L^2(\mathbb{R}^d)$ for all $n \geq 1$ be such that $\|f_n\|_2 \to \|f\|_2$ as $n \to \infty$. Suppose, moreover, that $$\int f_{n}g \to \int fg $$ ...
3
votes
2answers
86 views
Integral byFubini theorem
How can I solve this problem
$\def\R{\mathbb R}$
Suppose that $f$ is integrable on $\R^n$.
For each $t>0,$ let $E_t = \{x:|f(x)|>t\}$.
Prove $\int_{\R^n}|f(x)|dx = \int_0^\infty ...
3
votes
1answer
77 views
Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$
Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?
3
votes
1answer
53 views
Total variation
I cannot decide if the next function has bounded variation:
in the segment $(0,1)$
$$f(x)=\begin{cases}
\frac{1}{m^{2} n^{2}},&\text{if $x$ is rational}\\\\
0,&\text{otherwise}.
\end{cases}$$
...
3
votes
2answers
115 views
Lebesgue generalizations of Hilbert spaces?
Is an L[p] space a generalization of Hilbert spaces using Lebesgue integration?
And if this is the case, is it true that Holder's and Minkowski's Inequalities are generalizations of the ...
3
votes
2answers
235 views
Application of Radon Nikodym Theorem on Absolutely Continuous Measures
I have the following problem:
Show $\beta \ll \eta$ if and only if for every $\epsilon > 0 $ there exists a $\delta>0$ such that $\eta(E)<\delta$ implies $\beta(E)<\epsilon$.
For the ...
3
votes
1answer
39 views
Why $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?
How show that $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?
Can someone help me?
Thank you!
3
votes
1answer
41 views
Finiteness of measure
Let $\mu$ and $\nu$ be nonnegative Borel measures on $[0,+\infty)$ and let measure $\mu$ be finite, i.e. $\mu \{ [0,+\infty) \} < \infty$. For any $p > 0$ the equality holds
$$
...
3
votes
2answers
99 views
Showing that $\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$
How to show that:
$$\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$$
It seems like a easy example of illustrating 0 is not in the Lebesgue set of $g(x)$ where ...
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 ...
