1
vote
0answers
12 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
0
votes
1answer
35 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
3
votes
0answers
59 views

Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...
1
vote
0answers
37 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
1
vote
1answer
31 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
1
vote
0answers
29 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
2
votes
1answer
36 views

Inequality involving Jensen (Rudin's exercise)

Exercise (Rudin, R&CA, no. 3.25). Suppose $\mu$ is a positive measure on the space $X$ and let $f \colon X \to (0,+\infty)$ be such that $\int_X f \, d\mu=1$. Then for every $E \subset X$ ...
0
votes
0answers
65 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
2
votes
1answer
45 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
2
votes
1answer
39 views

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can ...
0
votes
1answer
47 views

Integration: Countable Additive Measure?

When considering Bochner's theory of integration one notices that having a countable additive measure rather than merely additive measure is not important, or do I miss something?
4
votes
1answer
59 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
-2
votes
2answers
82 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
2
votes
3answers
269 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
6
votes
2answers
115 views

Partial integration for lebesgue integrable functions

I want to show the following: Let be two Lebesgue integrable functions given: $f,g:[a,b] \rightarrow \mathbb R$. We define the functions: $$F,G: [a,b] \rightarrow \mathbb R : F(x)=\int_{[a,x]}^ \! ...
0
votes
0answers
48 views

Why is any continous function integrable?

He Everyone, For my Real Analysis course, I have a (probably very simple) problem, which I do not seem to get. Given: $f,g$ integrable functions with respect to measure $\lambda$ over the interval ...
3
votes
1answer
28 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
0
votes
1answer
70 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
1
vote
2answers
71 views

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions defined on a measure space $ (\Omega,\Sigma,\mu) $, where the index ...
1
vote
1answer
33 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
1
vote
0answers
20 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds ...
6
votes
3answers
111 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
1
vote
3answers
66 views

Showing that $\frac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is lebesgue integrable

I'm having some real trouble with lebesgue integration this evening and help is very much appreciated. I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over ...
2
votes
1answer
28 views

Showing function is lebesgue integrable

I have a function $f(x) = \frac{\sin(\frac{1}{x})}{1+(\log(x))^2}$ and I am trying to find whether this is Lebesgue integrable on $[1,\infty)$. I'm really not sure where to start on this one. It ...
4
votes
3answers
168 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
0
votes
0answers
42 views

Functions which can not be integrated via Riemann.

I am looking for some (possibly exotic) functions which can not be integrated via Riemann integration but can be integrated via Lebesgue. I am aware of the rational indicator function, i.e., $$ f(x) = ...
0
votes
2answers
23 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
2
votes
1answer
36 views

Reversing limits in Lebesgue integration

I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example, $$ ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
2
votes
1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
0answers
64 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
votes
1answer
33 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
0
votes
1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
2
votes
1answer
43 views

Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
3
votes
2answers
40 views

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as ...
3
votes
1answer
64 views

Inequality for integral => Inequality for integrand

I have that for any measurable set $\Omega\subset\mathbb{R}^d$ with $|\Omega|<\infty$ \begin{align}\sqrt{\int_\Omega f(x) dx }\leq \sqrt{c\cdot|\Omega|} + \sqrt{\int_\Omega g(x) dx }.\end{align} ...
0
votes
0answers
17 views

Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...
1
vote
0answers
25 views

a.e. convergence of dilations of a function in L^p

Let $f\in L^p(R^d)$ ($p<\infty$) and $\delta_h f(x):=h^{d/p}f(hx)$ (the normalization is so that $\|\delta_h f\|_{p} =\|f\|_p$). Consider $(h_n)$ a sequence of positive numbers such that either ...
2
votes
1answer
47 views

Lebesgue differentiation theorem with two variables

We know that if $f:\mathbb{R}\to\mathbb{R}$ is continuous then $$\lim_{h\to 0}\frac{1}{h}\int_x^{x+h} f(s)ds=f(x).$$ But if we have $f:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, what kind of ...
1
vote
0answers
22 views

Solve a problem of convergence of integral

We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$. How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial ...
2
votes
1answer
65 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
2
votes
1answer
51 views

Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
8
votes
2answers
405 views

Banach space valued integration (Riemann type)

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
3
votes
2answers
54 views

Limit of a certain Lebesgue integral

Can someone help me to show that $\lim_{t \to \infty} \int_{\Bbb R} f(x)\sin(xt)dx =0$ for any Lebesgue integrable function $f$? Side note: How do you make the $t \to \infty$ appear directly under ...
1
vote
0answers
29 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
1
vote
1answer
29 views

Reason for product sigma algebra notation

I was wondering why it is so common to denote the product sigma algebra with the same symbol that is used for tensor products. Is there a specific reason that this product symbol is used or was is ...
3
votes
1answer
34 views

What is the largest class of measurable functions $f$ s.t. $f'$ a.e.?

We know by Lebesgue Theorem that monotone functions on interval [a,b] has finite derivate almost everywhere and different of two monotone functions have finite derivative a.e. $\textbf{My Question}$ ...
0
votes
0answers
55 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
1
vote
2answers
77 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...