Tagged Questions

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.

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2
votes
1answer
37 views

Fubini's theorem for complete $\sigma$-algebras vs. non-complete $\sigma$-algebras

Suppose $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are both complete measure spaces. Consider the following two measure spaces: $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ and $(X ...
1
vote
2answers
40 views

Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Suppose $f$ is a real-valued (possibly infinite-valued) function on some measure space $(X, \Sigma, \mu)$, and suppose that it is measurable. Note that $f$ is not necessarily nonnegative. Does it ...
0
votes
1answer
32 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
2
votes
1answer
27 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
0
votes
1answer
31 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
votes
0answers
36 views

Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains an open interval [duplicate]

Suppose that $E$ and $F$ are Lebesgue measurable sets of $\mathbb{R}$, and their Lebesgue measures $m(E) > 0, m(F) > 0.$ Prove that $E + F = \{x + y : x \in E, y \in F \}$ contains a nonempty ...
0
votes
1answer
20 views

Lebesuge integrable function always bounded?

Assume $f: [0,1] \to \mathbb{R}$ is Lebesgue integrable, does it imply that $f$ is bounded almost surly?
3
votes
3answers
98 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
2
votes
0answers
38 views

McShane vs. Henstock-Kurzweil: Lebesgue integrable

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
1
vote
1answer
35 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
5
votes
1answer
157 views

Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = 0$

Let $f(x) \geq 0$ be continuous on the interval $[0, \infty)$, and suppose that $\int_0^\infty f(x)dx < \infty$. Prove that $\displaystyle \lim_{n \rightarrow \infty} \frac{1}{n}\int^n_0xf(x)dx = ...
2
votes
1answer
39 views

Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq \epsilon\}$ converges for each $\epsilon > 0.$

Let $\{f_n\}$ be a sequence of measurable functions on a measure space $(X, \mathcal{M}, \mu)$. Suppose that the infinite series $\displaystyle \sum_{n=1}^\infty \mu\{x \in X : |f_n(x)| \geq ...
2
votes
1answer
41 views

Is $\frac{\mathrm d}{\mathrm dx} \sin x/x = \cos x/x - \sin x/x^2$ Lebesgue integrable?

Is $$\frac{\mathrm d}{\mathrm dx} \frac{\sin x}{x} = \frac{\cos x}{x} - \frac{\sin x}{x^2}$$ Lebesgue integrable? In other words, is $$ \int_{\mathbb{R}} \left| \frac{\mathrm d}{\mathrm dx} ...
2
votes
1answer
46 views

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ ...
3
votes
2answers
155 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
1
vote
1answer
67 views

If the integral $\int_0^\infty xf(x)\,dx$ converges, so does $\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx$

Let be $f:[0, \infty)\rightarrow [0,\infty)$ a measurable function such that $$\int_0^{\infty}x\cdot f(x)\,dx< \infty.$$ Show that $$\sum_{n=1}^{\infty}\int_0^{\infty}f(x+n)\, dx<\infty .$$ ...
1
vote
0answers
36 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
1
vote
0answers
50 views

Lebesgue-integrable and existence of integral

I have given the following function $$ f(x,y) = \begin{cases} 1 &, \ x \in \mathbb{Q} \\ 2y & , \text{ otherwise} \end{cases} $$ This is a measurable function in sense of Lebesgue. Now, I ...
1
vote
1answer
40 views

Integral $ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $ (Lebesgue)

I have to compute the following integral: $$ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $$ $\chi_{B_k}(x) =\begin{cases} 1 &, \text{if } x \in B_k \\ 0 ...
2
votes
2answers
60 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
1answer
46 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
-1
votes
1answer
26 views

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$? What does it mean for $s$ to be integrable? 1. This is last minute exam revision. ...
1
vote
2answers
38 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
1
vote
1answer
65 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } ...
0
votes
1answer
29 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
0
votes
0answers
6 views

Images of intersections and Lebesgue integrals

How does the image not commuting with intersections, instead being just a subset of it $$f(U \cap V) \subset f(U) \cap f(V),$$ tell us that the Lebesgue integral is better than the Riemann integral? ...
0
votes
1answer
43 views

Absolute continuity and derivatives of integrals

I am preparing for a comprehensive at the end of the month, so I would appreciate any input I could get on this solution. I am pretty confident if the first part, but I think the second answer could ...
0
votes
0answers
47 views

Introduction of Lebesgue Theory

While reading Real Analysis- Stein, Shakarchi, I came up with following questions, some of which are in the "Introduction" of the book. The purpose of a series of questions is "to get actual ...
2
votes
2answers
121 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
0
votes
1answer
36 views

Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
0
votes
1answer
12 views

can representatives of an equivalence class of L_1(r^d)be nonmeasurable

Motivation: The composition of 2 Lebesgue measurable functions need not be measurable. This problem can be dealt with in a case by case basis( like with convolutions). Or as Big Rudin does, apply ...
3
votes
4answers
98 views

Can someone provide an example of Lebesgue integration on ordinary functions?

I'm trying to understand Lebesgue integral and integration but I'm completely mired by many author's motivation of this subject as only a tool for pathological functions such as the Dirichlet ...
2
votes
0answers
41 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
1
vote
0answers
20 views

Gauge Integral: Non-Borel Spaces

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 ...
1
vote
0answers
90 views

Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
0
votes
1answer
38 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
70 views

Gauge Integral: well defined?

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 ...
4
votes
2answers
134 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
0
votes
2answers
98 views

How to prove indicator function, step function, and then for sequences of step functions?

I try to proof a claim. It should be done first for indicator functions, then for step functions and finally for limits of increasing sequences of step functions. I'm not sure if I'm doing it right. ...
1
vote
2answers
153 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
4
votes
3answers
100 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
1
vote
0answers
43 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 ...
2
votes
0answers
25 views

Problem involving decomposition of measures

Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$ (We are given the following defintion: $\int{f} \> d\mu = \int{f} \> ...
1
vote
3answers
87 views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Suppose $(X, \mathfrak{A}, \mu)$ is a measure space. Let $\phi$ be a simple function with canonical representation $\sum^{k}_{n=1} a_{n} \chi_{E_{n}}$. I know we define the Lebesgue integral of $\phi$ ...
6
votes
1answer
52 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
1
vote
1answer
41 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 ...
0
votes
1answer
38 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
1
vote
0answers
35 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 ...
0
votes
0answers
43 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
3
votes
1answer
49 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 $E \subset X$ ...