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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45 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 ...
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28 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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19 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 ...
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41 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 ...
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26 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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30 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 ...
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30 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 ...
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1answer
47 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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62 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. ...
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1answer
34 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 ...
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2answers
47 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 ...
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35 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 ...
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1answer
39 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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26 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 $\{s(t)\}_{t\geq 0}$. ...
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106 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} ...
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1answer
56 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 ...
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67 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 ...
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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 ...
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52 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 ...
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1answer
15 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 ...
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3answers
49 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 ...
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55 views

Independent integrable random variables with 0 expectation so that $\overline{S}_n$ does not converge to 0 in probability

Give an example of independent integrable random variables $X_n$ such that $E[X_n] = 0$ for all n, but $\overline{S}_n = (\sum_{i=0}^n X_i)/n$ does not converge to 0 in probability. As far as I ...
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1answer
21 views

series of the integrals converges then the series converges almost surely

I know this was asked but I want a proof of this without using Fubini theorem. Anyway the first part of the problem can't be concluded using Fubini. I don't know how to do it :/ Let $f_k:\mathbb R ...
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26 views

Not sure if I understand the significance of support in these theorems.

I am just beginning to study the Lebesgue integral, and our building our way up to it. Right now we are defining the integral for bounded functions supported on a set of finite measure. In the ...
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1answer
78 views

$\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$ implies that random variables $X_n$ are uniformly integrable

$X_n$ are uniformly integrable if $\lim_{R \rightarrow \infty} \sup_n E[|X_n|,|X_n| \geq R] = 0$. Show that if $\sup_{n \geq 0} E[|X_n|\ln(|X_n|)] < \infty$, then $X_n$ are uniformly integrable. ...
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1answer
39 views

If $f$ is Lebesgue integrable on [0,1] show $g(x)=\int_{[x,1]} f(t)t^{-1}dt$ is Lebesgue integrable on [0,1]

Also want to show $\int_{[0,1]}g(x)dx = \int_{[0,1]}f(x)dx$. So since $f \in \mathcal{L}([0,1]), f=u-v$ where $u$ and $v$ are upper functions. Then I need to show $\int_{(x,1]} u(t)t^{-1}dt$ is an ...
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1answer
51 views

For a distribution function $F(x)$ and constant $a$, integral of $F(x + a) - F(x)$ is $a$.

For any distribution function and any $a \geq 0$, $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx = a$. In this case, "distribution function" means a right continuous function F with $F(-\infty) = 0$, ...
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1answer
26 views

Help understand the motivation behind this proof

The theorem states: for a function $f:X\rightarrow [0,\infty]$ that is measurable, if $$\int_E f\,\,d\mu=0$$Then, $f=0$ for almost everywhere on $E$. (Here $E\in\mathfrak M$, where $\mathfrak M$ is a ...
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1answer
20 views

Need correction for my “proofs” about the integrable functions.

Let $(X,\mathcal{A},\mu)$ be a measureable space, and assume that $\mu(X)<\infty$. Let $\left \{ u_{n} \right \}_{n\geq 1}$ be a sequence of functions in $\mathcal{L}^{1}(\mu)$ that converges ...
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0answers
27 views

proof in Holders inequality,(equality) [duplicate]

I have this proof in my book: I would like to prove what I underlined in red. but I get stuck. I guess in order to get equality we only need the opposite inequality. However I still don't ...
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1answer
31 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 ...
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26 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 ...
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1answer
38 views

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

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

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
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18 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 ...
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1answer
40 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 ...
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1answer
26 views

counter example related to Lebesgue integral

I am doing problems in Real Analysis by Stein, and was wondering if my solution to the following question is valid: I had to find a Lebesgue integrable continuous function whose limit superior ...
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42 views

How to prove $\int_{\mathbb R^d}{(1+|x|^{2})^{-m}dx}$ is integrable?

$\forall x \in \mathbb R^d$ and $m > d/2$. I know that $\int_{\mathbb R^d}{(1+|x|^{2})^{-m}dx}$ is integrable, but how can I prove? Thanks!
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1answer
58 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. ...
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1answer
72 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 ...
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1answer
79 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
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2answers
82 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 ...
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1answer
26 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 ...
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2answers
45 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 ...
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1answer
21 views

Adding integrals with different domains

Suppose I have two integrals $$ \int_{\Omega_1} f \, \, d \eta$$ and $$ \int_{\Omega_2} g \, \, d \eta$$ how would I define the sum of these two integrals? Is it possible? I want something of the ...
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1answer
41 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 ...
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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 ...
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2answers
77 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 ...
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2answers
74 views

What does $d\mathbb{P}(\omega)$ in integral mean?

What does $d\mathbb{P}(\omega)$ under integral sign mean? Like $$\int_B Xd\mathbb{\mathbb{P}}(\omega)$$ Can somebody explain? How can we integrate $X$ with respect to $\mathbb{P}(\omega)$ where ...
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22 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 ...