0
votes
1answer
25 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
0
votes
0answers
24 views

Determine integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$.

Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where ...
0
votes
1answer
30 views

Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
0
votes
0answers
15 views

regarding the well definedness of lebesgue integration's definition.

we know that the lebesgue integral for a simple function is defined by $$\int s=\sum_{n=1}^m\alpha_n\mu(s^{-1}\{\alpha\})$$. And we know that the canonical representation for a simple fuction need ...
1
vote
1answer
22 views

Proof of Egorov's theorem

I was wondering, looking at the proof of Egorove in :http://en.wikipedia.org/wiki/Egorov%27s_theorem, how could we be so sure as to say that each set $E_{n,k}=\bigcup_{m\geq k}\left\{x\in ...
2
votes
1answer
52 views

If $\mu(X)<\infty$, $\int |f_n|<C$ for all n, and $f_n\rightarrow f$ a.e.Show that $f_n\rightarrow f$ in $L^1$.

If $\mu(X)<\infty$, $\int |f_n|<C$ for all n, and $f_n\rightarrow f$ a.e.Show that $f_n\rightarrow f$ in $L^1$. I tried to use uniform integrability but I could not figure out completely. ...
3
votes
1answer
41 views

Properties of Lebesgue Integration

I am completely stuck with the following problem on Lebesgue Integration: Let $f:\mathbb{R}^d \to [0, +\infty]$ be measurable. Show that if $\int_{\mathbb{R}^d} f(x)dx < \infty$, then $f$ is ...
1
vote
2answers
68 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
1
vote
1answer
22 views

Does this proof for the MCT hold for the extended real valued functions.

Here is a proof for the MCT, but it says that it is for the real numbers, not the extended real numbers. If we allow the function f to take the value infinity does the proof still hold? I can not see ...
2
votes
2answers
36 views

Can simple functions take the value infinity?

I don't think my book is clear about this. It is "a course in real analysis", by weiss. Now I am in the chapter about the general lebesge integral, and we are going to develop the non-negative ...
1
vote
1answer
31 views

Can this exercise be solved by DCT, I was only able to use MCT.

How would you solve this exercise? You don't need to give me the details, just the general idea. Let f be a Lebesgue integrable function. Show that $\int f(x+a) d\lambda=f(x) d\lambda$ and ...
0
votes
0answers
22 views

Measure of triangular area

Let $\lambda\in[0,1]$, $\Omega=[0,1]^2$, $\vec{m}$ and $\vec{n}$ be two linearly independent vectors, $i\in\mathbb{N}$ and $h(t)$ the periodic extension of $$\tilde{h}(t):=\begin{cases} (1-\lambda)t ...
1
vote
2answers
73 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
0
votes
0answers
12 views

Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
1
vote
0answers
30 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
1
vote
0answers
33 views

Proof monotone convergence theorem, why do they use this lim sup?

I have a question about the proof of the MCT. First they use a lemma, this is ok, but I'll show it for completeness: Now comes the proof. But I am wondering, why do they use a lim sup here?, why ...
3
votes
1answer
69 views

Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Let $f(x)$ be a non-decreasing function on $[0, 1].$ You may assume that $f$ is differentiable almost everywhere. Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$. I am having a hard time with this ...
2
votes
1answer
39 views

Example with almost every convergence where the dominated convergence theorem fails

So I ran into this exercise, and I want someone to check the accuracy of my answer, because I feel pretty sure that I make some mistakes which I can't see. Let $f_n(x) : \mathbb{R} \to \mathbb{R}, ...
1
vote
1answer
25 views

(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...
3
votes
0answers
46 views

Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
6
votes
1answer
60 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
2
votes
1answer
36 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 ...
0
votes
1answer
28 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
26 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
29 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 ...
3
votes
3answers
97 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 $g \in L_{p'}(E)$, ...
1
vote
1answer
24 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
150 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
38 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
45 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 ∈ ...
1
vote
0answers
48 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
36 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
57 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
45 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
25 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
1answer
59 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
33 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 ...
2
votes
0answers
36 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
18 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 ...
1
vote
0answers
87 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
37 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
64 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 ...
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 ...
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 ...
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
81 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 $\phi$ ...
1
vote
1answer
36 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
0answers
39 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 ...