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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8
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501 views

Difference of differentiation under integral sign between Lebesgue and Riemann

Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
7
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0answers
269 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
6
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0answers
65 views

Various integration theories

Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely: Riemann Integration Riemann-Stieltjes Integration ...
5
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0answers
77 views

Theorem $2.14$ page $40, 41$ in Rudin - Real and Complex Analysis

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
5
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0answers
137 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
5
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0answers
1k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
4
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46 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
4
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0answers
96 views

$f(y-x)$ integrable implies $f=0$ a.e.

If $f(y-x)$ is in $L^p(\mathbb R^d\times\mathbb R^d)$, then I seem to conclude that $f=0$ a.e. (which seems wrong). My reasoning is that by Fubini and the integral's shift invariance (assume $p=1$ for ...
4
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175 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
3
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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 ...
3
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0answers
29 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
3
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0answers
96 views

Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
3
votes
0answers
36 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
3
votes
0answers
79 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
3
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0answers
86 views

Show: $\int f\, d\mu=\sum\limits_{y\in f(X)}y\cdot\mu(f^{-1}(y)))$

Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function which only takes a countable number of values. Show that $$ \int f\, d\mu=\sum\limits_{y\in ...
3
votes
0answers
675 views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
3
votes
0answers
96 views

How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space

Let $\mathcal{L}^2[(0,1)]$ denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1]. Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space. I believe that I can ...
3
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0answers
171 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
3
votes
0answers
341 views

Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and ...
3
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0answers
284 views

Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
2
votes
0answers
28 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 ...
2
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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} \> ...
2
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0answers
37 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
2
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0answers
48 views

Approximation of integration by simple functions.

Let $f: \Omega\longrightarrow \mathbb{R}$ be a Lebesgue integrable function. Does $$ s_n=\sum_{-\infty}^\infty\frac{k}{2^n}\lambda\left\{\frac{k}{2^n}<f\leq \frac{k+1}{2^n}\right\} $$ ...
2
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0answers
25 views

Question on Integrability of a function.

Let $(f_j)_{j\in\mathbb{N}}$ be a sequence of real functions in $L^p(X,\mathcal{A},\mu)$, where $p\geq1$. If we know that ...
2
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0answers
43 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
0answers
68 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
2
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0answers
90 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
2
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0answers
34 views

Measure Theory - Lebesgue Integral over non- $\sigma$-finite spaces

In most courses on Measure Theory the Lebesgue Integral is introduced initially for simple functions on finite spaces, then for general functions on finite spaces and finally for general functions on ...
2
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0answers
31 views

Proving this is Lebesgue integrable using radial functions

Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue ...
2
votes
0answers
47 views

Jump is no Lebesgue Point

Let $f$ be locally integrable, then for $x_0\in\mathbb{R}$ we have $$\lim\limits_{R\to 0}\frac{1}{|B_R(x_0)|}\int\limits_{B_R(x_0)}|f(x)-f(x_0)|dx=0.$$ The point $x_0$ is called Lebesgue point of $f$. ...
2
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0answers
192 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
2
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0answers
60 views

An application of Beppo Levi's Theorem

Can you help me solve this? Let $f$ be a continuous, monotonically increasing function on $\Bbb R$. Let $g$ be a non-negative measurable function over a measurable set $\Omega$. Prove that: ...
2
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0answers
43 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
2
votes
0answers
76 views

Lebesgue integral for a non-negative, measurable and bounded function

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f$ be a measurable, non-negative and bounded function. Show that the $\mu$-integral of f is given by $$ \int f\, ...
2
votes
0answers
103 views

Problem concerning Lebesgue Integral.

I am looking for feedback/corrections on the following solution attempt. I have only typed up one direction of the inequality; if this direction is correct, then I am certain that the other direction ...
2
votes
0answers
69 views

Does showing a function is integrable suffice to show the function is measurable?

I am reviewing past homework exercises in preparation for a midterm exam. Fortunately, my professor provides solutions. However, I found one of his solutions contains an (seemingly) important ...
2
votes
0answers
39 views

Integral of continuous function over probability distribution

Let $\mu_f$ denote the probability distribution of $f$ with $f(x)=10x-1$ for $x\in(0,1/2]$ and $f(x)=1$ for $x\in[1/2,1]$. If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, what is ...
2
votes
0answers
62 views

Prove that $\int_E cf(x)$ = $c \int_E f(x)$ for measurable E and non-negative f

So I'm going through class notes to try and prepare for an analysis qual. There are a number of little proofs in which the professor gave either no details or a brief sketch. I wanted to check if this ...
2
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0answers
70 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
2
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0answers
85 views

An almost orthogonality principle for $L^p$

If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
2
votes
0answers
323 views

solving this integral using Lebesgue dominated convergence theorem

I'm tried without sucess to solve this integral: $$\lim_{n\to\infty}\int_{0}^{1}(1+x^2)^{-n}dx$$ I know that the why to solve it ,is by using the Lebesgue dominated convergence theorem.
2
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0answers
152 views

Inner Approximation of Lebesgue Measurable Set

Can every bounded Lebesgue measurable set be approximated from the inside by countably many disjoint closed rectangles? A citation or proof would be nice.
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0answers
34 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 ...
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0answers
46 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 ...
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0answers
31 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 ...
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0answers
17 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 ...
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0answers
40 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
0answers
33 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 ...