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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0
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0answers
15 views

If $f$ is in $L^p$, prove that $\lim_{\lambda \to 0} \lambda ^p \omega(\lambda) = 0$

Suppose that $E \subseteq \mathbb{R}$ is measurable and that the measurable functions $f: E \to \mathbb{R}$ satisfies $\int_E |f| ^p < \infty$. If $\omega$ is the distribution function of $f$, ...
3
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2answers
41 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
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1answer
12 views

Derivative of Lebesgue integral function at the endpoints

Let $f$ be a non-decreasing Lebesgue-integrable real function on $[a,b]$. I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 340 here) that $$\lim_{h\to ...
0
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1answer
30 views

Measurability of derivative of Lebesgue integral function

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа that if $f:[a,b]\to\mathbb{R}$ is a Lebesgue-summable function on its domain then the derivative $\Phi'$ of the integral ...
1
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1answer
22 views

Definition of Summable

I am studying some material on real analysis. The word "summable" troubled me. I searched online. It looks like summable means having a Lebesgue integral. But Lebesgue integral is not a familiar ...
-1
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0answers
36 views

Differentiation under the integral sign for definite integrals [on hold]

Use Differentiation under the integral sign to compute $\displaystyle \int_0^\infty \frac{dx}{(x^2+p)^n}$ for any $n\in\mathbb{N}$. No idea even to start the solution. Can anyone please give me a ...
4
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1answer
30 views

Is there a simpler approach to this application of Dominated Convergence?

For a measure theory class, I'm trying to evaluate: $$\lim_{n\to\infty}\int^\infty_1\frac 1 {nx} e^{-x/n}\ \text d\lambda$$ Obviously I want to try and move the limit through the integral and ...
3
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1answer
30 views

Proving functions are in $L_1(\mu)$.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = ...
0
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1answer
37 views

Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
0
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1answer
42 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
0
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1answer
21 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
1
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0answers
21 views

approximation simple functions with finite support

Let $f$ be a nonnegative measurable function. I want to prove that there is an increasing sequence of nonnegative simple functions each of which vanishes outside a set of finite measure such that ...
1
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1answer
20 views

Union of $x$-sections measurable?

I know that the $y$-section $A_x$ of a $\mu_x\otimes \mu_y$-measurable set $A$, where $\mu_x\otimes \mu_y$ is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ (both measures being ...
0
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1answer
42 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
0
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1answer
22 views

Lebesgue integral involving distance function

Suppose $F$ is a closed set in $\mathbb{R}$, whose complement has finite measure, and let $\delta(x)$ denote the distance from $x$ to $F$, that is $$\delta(x)=d(x,F)=\inf\{ | x -y | : y \in F ...
0
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2answers
26 views

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
0
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0answers
40 views

Additivity of Lebesgue integral w.r.t. sets on non-finite domain

I know that for any Lebesgue integrable function $f:X\to\mathbb{C}$, or $f:X\to\mathbb{R}$, where $X$ is a set of finite measure such that $X=\bigcup_n A_n$, $\forall i\ne j\quad A_i\cap ...
0
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1answer
22 views

Integral of a product with any continuous function which has integral 0 is equal to 0

Let $g:[0,1]\to\mathbb{R}$ be bounded and measurable. For every continuous function $f$ with $\int_0^1f(x)dx=0$, $\int_0^1f(x)g(x)dx=0$ holds. I want to prove that $g$ is a constant function on ...
2
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1answer
30 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
1
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0answers
39 views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
0
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1answer
14 views

Approximate measurable function by simple function with compact support

Let $f$ be a nonnegative Lebesgue measure function on $\mathbb{R}$, $\epsilon>0$. How can we approximate $f$ by a nonnegative simple function $s$ with compact support s.t. $s\leq f$ and ...
2
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1answer
46 views
+50

Limit of Lebesgue integrable function

Let $f$ be a real valued, Lebesgue integrable function on $\mathbb{R}$. Prove that $$\lim_{t \to 0} \int_{\mathbb R} |f(x+t)-f(x)|\, dx=0.$$
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1answer
51 views

Monotone Convergence theorem

Give an example of a sequence of Lebesgue integrable functions $\{f_{n}\}$ converging everywhere to a Lebesgue integrable function $f$ such that $$\ \lim_{n \to \infty} \int_{-\infty}^{+\infty} f_{n} ...
0
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1answer
46 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
1
vote
1answer
27 views

Bounding a linear functional in $L_2[0, 1]$

For each f in $L_2[0, 1]$ let $\phi(t)$ be the solution of $y' + ay = f$ that satisfies $\phi(0) = 0$, where a is a constant. Define $l: L_2[0,1] \to \mathbb{C}$ by $l(f) = \int_0^1 \phi(t) dt.$ ...
0
votes
1answer
10 views

Euality between these two L1 integrable functions?

Let $$\lim_{n\to\infty}\int |u_n v_n-uv| d \mu=0$$ I want to show that $$\lim_{n\to\infty}\int u_n v_n d \mu=\int u v d \mu$$ What I have so far: $$\lim_{n\to\infty}\int u_n v_n d \mu-\int u v d ...
1
vote
1answer
48 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
1
vote
1answer
43 views

If $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$

If $X\geq 0$, and $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$. We know that that $$\mathbb{E}(\min(X,t))=\int_{X\leq ...
0
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1answer
48 views

If the Lebesgue integral of a strictly positive function is zero…

If the Lebesgue integral (over a set A) of a strictly positive function is zero, it means that the Lebesgue measure of A is zero? Thank you!
1
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1answer
36 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
0
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1answer
20 views

Exponential limit on sum of probabilities guarantees the product of powers of expectations is integrable

If X, Y are random variables and there exists a constant $c>0$ so that $P(|X| \geq x) + P(|Y| \geq x) \leq e^{-cx}$ for all x > 0, then $E[X^m Y^n]$ is integrable for all nonnegative integers m, ...
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0answers
17 views

Existence of finite Darboux sum with infinite partition

I would like to describe the class of all functions $a\in L^1(\mathbb{R},dx)$, such that there exists $\tilde{a}=a$ a.s. and a size $h$ of an infinite partition of $\mathbb{R}$, such that ...
0
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2answers
30 views

$\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$

I am trying to show the following: $\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$ I tried to use Fubini's theorem and the fact that ...
0
votes
1answer
23 views

Show the following Lebesgue integrals are equal.

Suppose that $f$ is integrable on $\mathbb{R}^d$. For each $\alpha >0$, let $E_{\alpha}= \{ x : |f(x)| > \alpha\}$ Prove that $$\int_{\mathbb{R}^d} |f(x)| dx = \int_{0}^{\infty} m(E_{\alpha}) ...
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0answers
28 views

Why the definition of the Lebesgue integrability is the finiteness of $\int\vert f\vert d\mu$?

I am studying the Lebesgue integration theory and I am encountered with the definition of the Lebesgue integrability. First, I will assume $f:X\to\mathbb{R}$ is a $\mathcal{A}$-measurable function ...
3
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1answer
25 views

Lebesgue integral of an improper Riemann integral

Let $f(x)= \frac{1}{\sqrt{x}}$ for $0 < x < 1$ . I am asked to show that for some enumeration on the rationals, $$F(x)= \sum_{n=1}^{\infty} 2^{-n} f(x - r_n)$$ is integrable. $\textbf{My ...
1
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1answer
44 views

Prove the following function is Lebesgue integrable.

Suppose $f$ is integrable on $[0,b]$. Show that $$ g(x)= \int_{x}^{b} \frac{f(t)}{t} dt$$ is integrable. $\textbf{My Attempt:}$ We want to show that $\int_{0}^b \mid g(x) \mid dx = \int_{0}^{b} ...
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0answers
28 views

Linearity of the integral without $\sigma$-additive measures

I was wondering how you could prove the linearity of the integral without using that measures are $\sigma$-additive. I have no clue of where to start, but let me state my question more precisely. ...
1
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1answer
31 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...
2
votes
2answers
37 views

Notation involving the Lebesgue integral.

I have a measurable function $f : \mathbb{R}^d \to \mathbb{R}$. Let $E$ be a measurable subset of $\mathbb{R}^d$. Then then $$\int_{E} f(x) \, dx = \int f(x) \chi_E (x) \, dx.$$ If we are taking an ...
0
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0answers
29 views

Integrability of a function from Stein Shakarchi Real Analysis

I have a question from Stein+ Shakarchi's Real Analysis book regarding the integrability of this particular function. (pg. 63-64) Consider the function $$f(x)= \begin{cases} \frac{1}{ \mid x \mid ...
3
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1answer
32 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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0answers
23 views

Existance of the integral in the domain of generator of the strongly continuous semigroup

Let $\{s(t)\}_{t\geq 0}$ is a $C_0$ semigroup of bounded operator on the Banach space $X$ and $A:D(A)\subset X\rightarrow X$ be the infinitesimal generators of the semigroup $\{s(t)\}_{t\geq 0}$. ...
9
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1answer
100 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
0
votes
1answer
48 views

Real Analysis - Lebesgue integrable functions

Let $E$ be a measurable set. Suppose $f \geq 0$ and let $E_k=\{x \in E_k|f(x) \in (2^k, 2^{k+1}] \} $ for any integer $k$. If $f$ is finite almost everywhere, then $\bigcup E_k = \{x \in E |f(x)>0 ...
3
votes
1answer
49 views

Counterexample to "if $\int_E f < \infty$, then $\lim_{n \to \infty} \int_A f_n = \int_A f$

Part a) of the question is as follows: "Suppose that $E \subset \mathbb{R}^d$ is a measurable set and that $f, f_n$ are measurable functions on $E$ satisfying $f_n \to f$ a. e. on $E$. Suppose that ...
2
votes
0answers
38 views

Is it possible to abstract a Riemann integral into a “higher” integral with measure?

I'm not very comfortable with more generalised integrals such as the Lebesgue integral yet, but I'm working through some material to achieve that goal. I have a question which stems simply from ...
3
votes
0answers
41 views

Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
0
votes
1answer
13 views

Finding a bound for the maximum function

the following problem says: Show that if is f an integrable function in $\mathbb{R}^d$ and not identically null, then $$f^*(x)\geq\frac{c}{|x|^d}$$ where $c>0$, $|x|\geq 1$ and $f^*(x)=\sup_{x\in ...
0
votes
3answers
46 views

Complex Lebesgue integral, property

Lets say that you for real functions have proved that: $|\int_{\Omega}fd\mu|\le \int_{\Omega}|f|d\mu$. How do I then prove that it also holds for complex-valued functions? I guess this amounts to ...