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.

learn more… | top users | synonyms

1
vote
1answer
29 views

$L^2$ and $L^1$ space problem

For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
5
votes
0answers
50 views

Weakly convergence

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and we define a function $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ a bounded ...
-3
votes
1answer
37 views

How can I solve like this exercise in measure theory [on hold]

If $J=\{[a,b[$ : $a \le b$ : $a,b \in R\}$ and $F$ is an continous increasing bounded function on $R$ , and if we put $λ([a,b[)=F(b)-F(a)$ prove that : $$λ(\emptyset) = 0 $$ and if the union of ...
0
votes
2answers
104 views

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?

Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why and why isn't this true for $1/x$? I'm having difficulty understanding difference between the above functions in terms of ...
1
vote
1answer
33 views

Confused about switching Lebesgue integrals for Riemann integrals

Hi I have been attempting given in the link below. I am confused about the argument used to show the function is not Lebesgue integrable. This question What each person has used to answer is the ...
3
votes
1answer
39 views

Convergence in $L^1_{loc}$ implies convergence almost everywhere

Let $f_n\in L^1_{loc}(\mathbb{R})$ be a sequence of a locally integrable functions such that for all $a<b$ $$\int_a^b|f_n(x)|dx\to 0,$$ when $n\to\infty$. We know that for each interval $[a,b]$ ...
3
votes
2answers
40 views

Proving that $m(E) = 0$ if for all $n$, $\int_E x^n \cos x\, d x = 0$

Suppose that $E\subset [0,2\pi]$ is measurable and $\int_E x^n \cos x\,dx = 0$ for all $n =0,1,2,\cdots$. Then prove that $m(E)=0$. In a non-rigorous fashion, if $\sum_{1}^{\infty} a_nx^n = \sec ...
0
votes
1answer
50 views

Is a compact set an union of a finite number of disjoint closed intervals?

I think it is true for $\mathbb R$ with usual metric. How about others? How to prove it? Motivation: I got this idea when I was reading a proof for Lebesgue's Criterion of Riemann Integrability, here ...
2
votes
0answers
28 views

If a simple function is measurable, then is its characteristic function

I am working on a problem looks like this: If a simple function $s$ is measurable, show that its characteristic function $\mathcal X_{X_i}$ is measurable. Here are the ways I have been working ...
1
vote
2answers
76 views

Lebesgue Integral: $\int_1^{\infty}\frac{1}{x}$

The following is an exercise from Carothers' Real Analysis: Show that $$\int_{1}^{\infty}\frac{1}{x}=\infty$$ (as a Lebesgue Integral). Attempt: Let $E=[1,\infty)$. $\int_E f=\int f\cdot ...
0
votes
0answers
21 views

A question about countably subadditive property of Lebesgue Outer Measure

Here is the definition of Carothers' Lebesgue Outer Measure: . And countably subadditive property of Lebesgue Outer Measure has been talked here: I can understand all proofs. However, I'm ...
0
votes
1answer
34 views

How will m*(rE) behave?

Let $rE =\{rx: x\in E\}$, what is $m^*(rE)$ in terms of $m^*(E)$? Intuitively, I think $m^*(rE)\leq r\times m^*(E)$. However I've no idea how to prove it? Add: Definition of Lebesgue Outer Measure ...
1
vote
1answer
23 views

Disjoint convex sets which cannot be separated by any continuous linear functional

This problem is out of Rudin's Functional analysis exercise 3.2. The problem is stated below. I'm really struggling with this chapter in general. It has a lot of new topics I have not seen before. Any ...
0
votes
0answers
18 views

Conditions for 2 variable functions to be Lebesgue integrable

I'm trying to solve this problem but am having some issues. So I understand the conditions required to show a 1 variable function is integrable on some $E$, a subset of $\mathbb{R}$. But, if the ...
0
votes
0answers
37 views

Integral of limit of a function

I am working on a problem$^{(1)}$ similar to this 2013 posting: Suppose that $f_n$ is a sequence of integrable, non-negative functions, so that $\forall x$, $f_n(x)$ decreases to $f(x)$. Show the ...
0
votes
1answer
55 views

How to explain the why here?

Reading Lebesgue outer measure of Lebesgue Measure Chapter from Carothers' Real Analysis and some properties and their proofs are here: Basically, I can't understand the proof for reverse ...
1
vote
2answers
41 views

On continuous function of compact support

Hello all I am stuck on the following small question in real analysis for practice in which we are given a function f of compact support and a measurable set A we are asked to prove the following is ...
0
votes
2answers
29 views

Countable additivity of Lebesgue integrals proof

Show if $f_n$ are non negative measurable functions: $$\int (\sum_{n=1}^\infty f_n) d\mu = \sum_{n=1}^\infty \int f_n d\mu$$ Does this not just follow from the theorem for two additivity? Say $\int ...
1
vote
1answer
40 views

Showing that $\int_{E\cup F}f=\int_E f+\int _F f$, where $E\cap F=\emptyset$

I would like to show that $\int_{E\cup F}f=\int_E f+\int _F f$, where $E\cap F=\emptyset$ and $E,F$ are Lebesgue measurable sets. Attempt: First I tried to show that in general I can write $\int ...
0
votes
0answers
55 views

Question in real analysis and Lebesgue integration

So yeah my school makes us take a mini real analysis course for physics and I am really stumped on this one: I have the following question and would certainly appreciate any help please : we are given ...
1
vote
1answer
27 views

Question in real analysis an Minkowski difference

I have the following question in real analysis: I was first asked to prove that if the function which I proved to be continuous has a point x such that F(x) > 0 then there exists an open cube in ...
3
votes
2answers
46 views

Use the Lebesgue's Monotone Convergence Theorem and the fact that $\int^\infty_0t^ne^{-t}dt=n! \text{ to prove that }\int^\infty_0f(t)e^{-t}dt=s.$

Consider a sequence of real numbers $(a_n$ with $n\in\mathbb{N}_0$, such that $a)n \geq0\text { non-negative}$ for all $n\in\mathbb{N}_0$ and $\sum^\infty_{n=0 } =s\in\mathbb{R}$ b. Use the ...
2
votes
0answers
33 views

Question about definition of the Lebesgue integral of a non-negative function

I am reading Royden's Real Analysis to learn about Lebesgue integration. Royden first shows that a bounded function on a set of finite measure is Lebesgue integrable if and only if it is measurable. ...
3
votes
1answer
36 views

A rigorous meaning of “induced measure”?

In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = ...
1
vote
3answers
62 views

Integration of $\int_{0}^{1} x^{a}(1-x)^{-1}\log (x) dx $

I need to compute the following integral for $a>-1$, $$\int_{0}^{1} x^{a}(1-x)^{-1}\log (x) dx $$ My attempt: By change of variable $x=1+t$: \begin{align*} I &= \int_{0}^{1} ...
1
vote
2answers
61 views

integration of $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} \frac{(-1)^n x^{2n}}{4^n (n!)^2}dx $

I need to compute the following integral for $a>1$ $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} \frac{(-1)^n x^{2n}}{4^n (n!)^2}dx $ My attempt $\int_{0}^{\infty} e^{-ax}\Sigma_{0}^{\infty} ...
2
votes
1answer
25 views

$x^{-1}\sin x$ is not integrable on $[0,\infty)$ implies $e^{-xy}\sin x$ is not integrable on $[0,\infty) \times [0,\infty)$

I am reviewing a homework problem and I came upon the following statement, which is only a part of what I am trying to solve: Given that $\frac{\sin x}{x}$ is not Lebesgue integrable on ...
1
vote
1answer
41 views

Integration on manifolds and improper integration

Consider the usual concept of integral on a smooth manifold (the one built using partitions of unity). When applied to the usual smooth structure of $\mathbb{R}^n$, does it coincide with the concept ...
1
vote
1answer
18 views

$g$ is differentiable and $g'(y)=\int_{\mathbb{R}}ixf(x)e^{iyx}dm(x)$

Let $f \in \mathcal{L}(\mathbb{R},\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ measurable Lebesgue. Asumme that $x\to f(x)$ is measurable. For $y \in \mathbb{R}$ define: ...
0
votes
2answers
38 views

$\int_{0}^{\infty} x^{a-1}(e^x-1)^{-1}dx $

I need to show that I= $\int_{0}^{\infty} x^{a-1}(e^x-1)^{-1}dx = \Gamma(a) \times \Sigma n^{-a}$ where $a > 1$ I have no clue how to approach ! I am using $\Gamma(a) = \int_{0}^{\infty} ...
2
votes
0answers
15 views

Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi d\lambda$; where ...
2
votes
1answer
24 views

Lebesgue's differentiation theorem for all points

Let $f \in L^2(0,T)$ be such that $f(t)$ is well-defined for every $t$ (not just a.e. $t$). But I have no continuity of $f$. We have by Lebesgue's differentiation theorem that $$\lim_{a \to ...
0
votes
1answer
48 views

Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive?

Suppose $f$ is measurable. Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive? If we require $f(x) \geq 0$, the equality holds by Lebesgue's ...
0
votes
1answer
27 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
7
votes
0answers
115 views

Is Stokes theorem on $n$-cells equivalent to Stokes theorem with singularities

Could you tell me if Stokes theorem for $n$- cells is equivalent to Stokes theorem for manifolds with singularities (where the set of singularities has measure zero/ is negligible)? I mean, would it ...
2
votes
1answer
65 views

Proof that for the Lebesgue indefinite integral, $\int_E fd\mu(x)=0$ implies $f=0$ almost everywhere

Can someone provide a hint for the proof of the fact that for the Lebesgue indefinite integral, $\int_E fd\mu(x)=0$ for all $E\in S$ where $S$ is a the $\sigma$ ring, implies $f=0$ almost everywhere? ...
0
votes
1answer
39 views

Inequality for convolution $| f *g| \le \|f\|_1 \|g\|_1$ [closed]

Let $f \in L^1$ and $g \in L^1 \cap C^{\infty}_c, $ is it then true that a.e. we have $$ |f *g| \le \|f\|_1 \|g\|_1?$$
3
votes
0answers
55 views

Convergence of functions in $L^1$ implies convergence of derivatives $ a.e. $?

Update: Someone gives me a good counterexample, which basically answers all the questions I posed. The example is, $$f_n(x)=\frac{sin(nx)}{n}$$ ----------------------------------------- I just came ...
0
votes
0answers
36 views

a function that is not measurable but its square is measurable

Question: Give an example of a function $f$ on $X$ to $R$ which is not $X$ measurable, but is such that the functions $|f|$ and $f^2$ are measurable. To me, $|f|$ is measurable and $f^2$ is ...
2
votes
1answer
39 views

Problem about limit of Lebesgue integral over a measurable set

This is actually problem 4T of Bartle's book "The elements of integration and Lebesgue measure". Let $f_n$, $f$ be nonnegative measurable functions on $\mathbb{R}$ such that $f_n\to\ f$ for every ...
0
votes
3answers
54 views

If $f \in L^{1}(X,\mu,\mathfrak{M},\mathbb{R})$ , $\int_{E_n}f \to \int_{X}f $ [closed]

Let $f \in L^{1}(X,\mu,\mathfrak{M},\mathbb{R})$ where $\mathfrak{M}$ is the $\sigma$-algebra of Lebesgue and $\mu$ measurable Lebesgue. If $E_1\subseteq E_2\subseteq E_3 \subseteq E_4\subseteq ...
2
votes
1answer
25 views

Exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$.

For all $f:[0, 1]\to \mathbb{R}$ exist $g:\mathbb{R}\to \mathbb{R}$ Lebesgue measurable and $h:[0, 1]\to \mathbb{R}$ Borel measurable such that $f = g \circ{}h$. Any ideas. Thanks
1
vote
3answers
37 views

If $|\int f_n-\int f |\to 0 \Rightarrow\int |f_n - f| \to 0$.? [closed]

If $|\int f_n -\int f| \to 0 \Leftrightarrow |\int (f_n-f) |\to 0$ where $f_n$ and $f$ have the hypothesis of dominated convergence theorem It is true that: If $|\int f_n-\int f |\to 0 ...
2
votes
1answer
36 views

Proof of a double series equality

Let $b_{n_{i}} \in \mathbb{C}$ for $n,i\in \mathbb{N}$. Suppose that $$\sum_{n=1}^{\infty}\sum_{i=1}^{\infty}|b_{n_i}|<\infty,$$ then ...
0
votes
0answers
24 views

Is this Integral transformation correct?

I have an Integral: $$ \int_{-\infty}^{-y_1} \Phi(y_2)d\Phi(x_1) $$ Here: $\Phi(y_2)$ is the Gaussian density function of variable '$y_2$' which has to be integrated w.r.t Gaussian density of ...
1
vote
1answer
21 views

$f$ be Lebesgue integrable and $F = m\{f > \alpha\}$, then $F$ is right continuous

The following is a part of a problem 18.2 from from Real Analysis, N. L. Carothers: Let $f: \mathbb{R} \rightarrow [0, \infty]$ be integrable and define $F: [0, \infty) \rightarrow [0,\infty]$ by ...
2
votes
1answer
24 views

Showing $\left(\sum_1^\infty 2^{-n} \mathbf{1}_{x\in(0,1)}(x-r_n)^{-1/2}\right)^2$ is not integrable

A small part to a homework problem. Let $\{r_n\}_1^\infty$ be an enumeration of the rationals, and let $f(x) = x^{-1/2}$ if $0<x<1$ and $f(x) = 0$ otherwise. Define $g(x) = \sum_{n=1}^\infty ...
1
vote
0answers
28 views

Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
0
votes
0answers
20 views

Differentiating the expectated area under stochastic process

I am trying to prove the following, where $X_t$ is an almost surely bounded progressivley measurable process: $$ \lim_{t\rightarrow 0}\frac 1 t \mathbb E \int_0^t X_sds = \lim_{t\rightarrow 0} ...
3
votes
2answers
19 views

Measurable functions such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$.

I need to find an example of a suquence of measurable functions $ f_n \geq0$ for $ n = 1,2, ... $ such that $\int\underline{\text{lim}}f_n=0 $ and $\underline{\text{lim}}\int f_n=+\infty$. As I can ...