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.
3
votes
1answer
92 views
How show that $\lim_{\varepsilon \rightarrow 0}\int_A h_\varepsilon(x)dx =0$, whenever $\bigg|\int_I h\bigg|\leq |I|^{1/2}$?
Let $h$ be a bounded, measurable function, such that, for any interval $I$
$$\bigg|\int_I h\bigg|\leq |I|^{1/2}.$$
I want show that, for any $A$, with $|A|<\infty$,
$$\int_A ...
2
votes
1answer
51 views
Interchange differential operator with Lebesgue integral.
Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
2
votes
1answer
34 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
2
votes
1answer
47 views
Convergence and Lebesgue Integration
I came across this question in a textbook on introductory Lebesgue Integration. I have been teaching myself this material but was unsure of how to do the following question:
Let $(g_n)$ be a sequence ...
2
votes
1answer
108 views
Riemann-Stieltjes integrability criterion
I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7:
Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
2
votes
1answer
71 views
A question about Lebesgue integral.
Let $f$ be Lebesgue measurable, and $\int_a^b x^\alpha f(x) = 0$ for every $\alpha\ge 0$.
How do I show that $f(x)=0 ~ a.e.$
and if the condition change "$\alpha\ge 0$" to "$\alpha\ge k$ for some ...
2
votes
1answer
335 views
Lebesgue integrable function?
If $\displaystyle f(x)=\frac{1}{x^p}$ $(0 < x \leq 1)$ then $f \in L[0,1]$ if $p<1$ and
$$\int_{0}^1 f= \frac{1}{1+p} $$
I know that non negative measurable function f is Lebesgue integrable ...
2
votes
1answer
280 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| = ...
2
votes
1answer
97 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 ...
1
vote
1answer
39 views
Prove $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have $\lim\limits_{x\to \infty} F(x)=0$.
$f$ is integrable on $[0,\infty)$, and $\int_0^{\infty} |f(y)|dy < \infty$.
Prove:
Then $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have ...
1
vote
1answer
34 views
Bounded integrable function
Let $f : \mathbb{R} \to \overline{\mathbb{R}}$ be an integrable funtion. Given $\varepsilon > 0$ show that there is a bounded integrable function $g$ such that $\int |f - g| < \varepsilon$.
I ...
1
vote
1answer
63 views
Typical problem in functional analysis
I need help with this problem from my homework
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
1
vote
1answer
60 views
Riemann-Lebesgue equivalence for n-dimensional integration
"Lebesgue's Theorem" states that for any bounded $f:[a,b] \to \mathbb{R}$, $f$ is Riemann Integrable iff $m\{x:f \text{ is not continuous at x }\}=0$, and if so Riemann's integral coincides with ...
8
votes
0answers
185 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
0answers
55 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
...
4
votes
0answers
100 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
496 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
0answers
210 views
Riemann integral vs Lebesgue integral
Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between
$$ \int_{D(0,R)}|f(w)|dm(w)$$
and
...
3
votes
0answers
107 views
How to apply Borel-Cantelli Lemma?
Assume that we are given a sequence of continuous functions $f_n(x)$ on $[0,1]$.
How to show the existence of a sequence $a_n$ and a set $A$ with $\mu(A^c)=0$ so that
$$
\lim_{ n\to \infty} ...
3
votes
0answers
122 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 ...
2
votes
0answers
32 views
Question about Lebesgue integration on $\mathbb{R}^N$
Let $\Omega\subseteq\mathbb{R}^N$ be an open set and $f:\Omega\to[0,+\infty[$ a measurable function, bounded over each compact $K\subset\Omega$. If there is a $C>0$ such that
...
2
votes
0answers
51 views
How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space
Let $\mathcal{L}^2[(0,1)]$
denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1].
Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space.
I believe that I can ...
2
votes
0answers
63 views
Prove Heisenberg uncertainty principle (measure and integration theory)
Here is a question in measure and integration theory,
Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
2
votes
0answers
64 views
Riemann Stieltjes integral definition and implications
I am studying the Riemann Stieltjes on Tom Apostol's book mathematical analysis second edition and I have a the following question.
Given $[a,b]$ we define a partition of this interval to be a set $P ...
2
votes
0answers
132 views
solving this integral using Lebesgue dominated convergence theorem
I'm tried without sucess to solve this integral:
$$\lim_{n\to\infty}\int_{0}^{1}(1+x^2)^{-n}dx$$
I know that the why to solve it ,is by using the Lebesgue dominated convergence theorem.
1
vote
0answers
57 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
vote
0answers
29 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
1
vote
0answers
29 views
The tightest bound on an integral
Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
1
vote
0answers
33 views
continuous on Lebesgue space
$f\in \mathcal{L}^{\infty}(R), t>0,x \in R, B(x,t)=\{ y\in R: |x-y|<t \}$, and $u(x,t)$ is defined as:
$$ u(x,t)= \frac{1}{2t} \int_{B(x,t)}e^{ity}f(y)dy $$
(1) $u(x,t)$ is continuous on ...
1
vote
0answers
75 views
Are continuous functions strongly measurable?
Measure theory is still quite new to me, and I'm a bit confused about the following.
Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
1
vote
0answers
39 views
Is the Lebesgue integral essentially an inner measure of some kind?
Consider a non-negative Lebesgue-integrable function $f : X \rightarrow \mathbb{R}$, where $X$ is a measure space, and let $F = \{(x,y)|x \in X, y \in [0,f(x)]\}$. Can the Lebesgue integral of $f$ be ...
1
vote
0answers
83 views
Characterization of Dirac Measure
Let $x_0$ be a point in a set $X$ and $\delta_{x_0}$ the Dirac measure concentrated at $x_0$. Characterize the nonnegative real-valued functions on $X$ that are integrable over $X$ with respect to ...
1
vote
0answers
71 views
Uniform convergence in $L^p$-spaces
Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$.
...
1
vote
0answers
51 views
Infinite function on measure zero set
I am trying to prove that given a set $E$ of measure 0, and a function $f \equiv \infty$ on $E$, then $\displaystyle \int_{E}f = 0$.
This would be easy if one is allowed to assume that $\infty \times ...
1
vote
0answers
88 views
Inner Approximation of Lebesgue Measurable Set
Can every bounded Lebesgue measurable set be approximated from the inside by countably many disjoint closed rectangles?
A citation or proof would be nice.
1
vote
0answers
45 views
Lebesgue-Stieltjes integral as a generalized function
Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral
$$
\langle F, \varphi \rangle ...
1
vote
0answers
67 views
Lebesgue measure is invariant under isometry
Is it true that Lebesgue measure is invariant under isometric map? I mean standard measure of $R^n$.
It is certainly true for interval in $R$ (obvious). I've attempted to prove it in general by ...
1
vote
0answers
65 views
About Lebesgue measure
This is a problem of Lebesgue measure and measure theory specifically.
Suppose that
$f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable.
$\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue ...
1
vote
0answers
70 views
$\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm$
Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that
$$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$
Proposition: $f$ is ...
1
vote
0answers
187 views
Extended Riemann integrability of a non-negative function implies Lebesgue integrability?
Let $f$ be a bounded function on a finite interval $[a, b]$ of the real line.
If $f$ is Riemann integrable, we denote its Riemann integral by $\mathcal{R}(f , [a, b])$.
It is well known that $f$ is ...
1
vote
0answers
262 views
showing that a sequence is uniformly integrable
I am currently reading the new edition of Royden and I've gotten to a part where the book made some comments without justification and I'm trying to verify these facts on my own. I want your help in ...
0
votes
0answers
45 views
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
Show that
$\int_{0}^{1} dx \int_{0}^{1} f(x,y) dy=\frac{\pi}{4}$
$\int_{0}^{1} dy \int_{0}^{1} f(x,y) dx=-\frac{\pi}{4}$
0
votes
0answers
24 views
Lebesgue integrable functions
Suppose
$$f(x) = \sum_{n=0}^{\infty} \frac{1}{2^n} \phi(2^n x)$$ on the closed interval $[0,1]$ where $\phi$ is given by
$$\phi(x) = \begin{cases} x, & \mbox{if } x \in \left[0,\frac12\right] \\ ...
0
votes
0answers
70 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
0
votes
0answers
54 views
Change of differentation and integration signs.
I'm going through an old exam in a course I'm taking. I have the given rule:
Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
0
votes
0answers
48 views
Supremum of norms of line integrals
I have the following problem:
Let's say $\Omega\subset\mathbb{R}^2$ is a bounded open connected set with Lipschitz boundary and $f\colon\Omega\rightarrow\mathbb{R}$ is a function such that $f\in ...
0
votes
0answers
33 views
Is it possible to show that if $-f , f \in L^{+}(I)$ then $f\in R(I)$
Is it possible to show that if $-f , f \in L^{+}(I)$ then $f\in R(I)$?
We can rewrite the integrals of $-f$ and $f$ on $I=[a,b]$ as:
$$\int_{a}^{b} fdx = \lim_{n\rightarrow ...
0
votes
0answers
61 views
Question about an integration method in Analysis
I have a question about an integration method widely used in Analysis, namely the fact that
$$
\int_{B(x_0,R)}
{
\hspace{-20pt}
f(x)\,{\rm d} x
}
=
\int_0^R
{
\hspace{-5pt}
...
0
votes
0answers
155 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 ...
-4
votes
0answers
34 views
Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I. Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I need to solve the below.
(1)Show $\mu(\{x\in X| ...
