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
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1answer
18 views

Sufficient conditions for the Hardy-Littlewood Maximal function $M(f)$ being continuous

There are four common versions of Hardy-Littlewood Maximal operator $M(f)$: centered/uncentered + ball/cube. I noticed that the continuity of $M(f)$ depends on the version. For example, let $f$ be the ...
1
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1answer
43 views

Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Let $n \geqslant 1$, Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$, where $u \in (0,1)$. What I did is just use the definition of ...
1
vote
1answer
18 views

Is this theorem an extension of Scheffé Lemma

Let $\nu_n(A)=\int_A f_n d\mu$ and $\nu(A)=\int_A f d\mu$ where $f_n, f$ are density functions. If $\nu_n(X)=\nu(X)$ and $f_n\to f$ ($\mu$- a.e) then $$\sup\limits_{A\in \mathcal ...
1
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0answers
45 views

If $\nu (A)=\mu(A\cap B)$ then $\nu \ll \mu$

If $\mu$ and $\nu$ are two measures on the same measurable space then $\mu$ is said to be absolutely continuous with respect to $\nu$, or dominated by $\nu$ if $\mu(A) = 0$ for every set $A$ for which ...
3
votes
3answers
79 views

If $\int_A f d\mu =\int_A g d\mu$ for $\forall A\in \mathcal A$ and $\mu$ is $\sigma$-finite then $f=g$ a.e

Functions $f,g$ are nonnegative on $(X,\mathcal A, \mu)$. If $\int_A f d\mu =\int_A g d\mu$ for $\forall A\in \mathcal A$ and $\mu$ is $\sigma$-finite then $f=g$ a.e Can I discard the condition ...
1
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1answer
43 views

Question on Lebesgue integration and continuity

$(X,\mathcal A, \mu)$ is measurable space. Suppose that function $f(t,x)$ is measurable w.r.t variable $x$ for each $t\in (a,b)$. a) $f(t,x)$ is continuous at $t_0$ for all $x\in X$, and ...
1
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1answer
53 views

Show that the Fubini Tonelli theorem does not work for this function

Problem Statement: Let $X = Y = R$ and let $B$ be the Borel $\sigma$-algebra. Define $$ f(x,y) = \left\{ {\begin{array}{*{20}{c}} 1&{x \ge 0{\text{ and }}x \le y < x + 1}\\ { - 1,}&{x \ge ...
0
votes
1answer
27 views

If sum of a set of increasing functions defined on $[a,b]$ is convergent, will this sum be Lebesgue measurable?

If $\{f_n\}$ is a set of increasing functions defined on $[a,b] \subset \mathbb R$ and $\sum_{n=1}^{+\infty} f_n$ converges to $F(x)$, will $\sum_{n=1}^{+\infty} f_n$ be Lebesgue measurable? I got ...
5
votes
2answers
46 views

Question on Lebesgue point integration

If $f(x)$ is finite at $x$ and $\lim\limits_{h\to 0}\frac{1}{h}\int_x^{x+h} |f(t)-f(x)|dt = 0$ then $x$ is called a Lebesgue point of function $f$. a) If $f$ is continuous at $x$ then $x$ is a ...
1
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1answer
22 views

“$f\ge 0$, $g>0$, $fg\in L^1(\mathbb{R})$ and $g\notin L^1(\mathbb{R})$” implies “$f$ is integrable over $[-r,r]$”?

Let $f\ge 0$ and $g>0$ be such that $fg\in L^1(\mathbb{R})$ but $g\notin L^1(\mathbb{R})$. Can we get the following conclusion: $f$ is integrable over $[-r,r]$ for any $r>0$ ? Intuitively I ...
1
vote
1answer
64 views

If $\int_A f d\mu \leq \int_A g d\mu$ for all $A$ then $f\leq g$ a.e

$f, g$ are nonnegative functions on $(X,\mathcal A, \mu)$ and $\int_A fd\mu \leq \int_A g d \mu, \forall A \in \mathcal A$. Show that if $\{f>0; g< \infty\}$ is a $\sigma$-finite set then ...
4
votes
1answer
30 views

Question on Lebesgue integration and partition?

$A_1, A_2, \ldots, A_n$ are measurable sets w.r.t $(X,\mathcal A,\mu)$ with $\mu(X)<\infty$. If each $x\in X$ belongs to at least $q$ sets in $A_1, A_2, \ldots, A_n$ then there exits a set $A_i$ ...
2
votes
2answers
50 views

If $f$ is Lebesgue integrable on $ R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0? $

If $f$ is Lebesgue integrable on $R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0$ My attempt: I am trying to use the definition of Lebesgue integration but I am stuck. Can anyone give me some ...
5
votes
1answer
41 views

For Lebesgue integrable $f$, show $\lim_{t\to +\infty}t\mu(\{x : |f(x)| \ge t \})=0$

Function $f$ is integrable on set $A$ w.r.t measure $\mu$ and $A_t=\{x\in A: |f(x)|\geq t\}$. Show that $\lim_{t\to \infty} t\mu(A_t)=0.$ I think that question is false, should $\lim_{t\to ...
6
votes
3answers
91 views

Question on Lebesgue integration?

Define $$\int_{E^*}f\ \mathsf d\mu = \inf_{\{A_i\}} \sum_i\left[\sup\limits_{x\in A_i}f(x)\right]\mu(A_i),$$ where $\{A_i\}$ is a partition of $E$. I want to prove that if $$\mu(\{x: f(x)>0\}) ...
1
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0answers
47 views

Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

For what nonnegative integer $n$ and positive real $c$ does the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral and when does it converge as an ...
2
votes
2answers
79 views

Prove that $\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $

Suppose $f_n$ are nonnegative measurable functions on a measure space satisfying $$\int f_n\ \mathsf d\mu =1.$$ Prove that $$\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $$ almost ...
2
votes
1answer
46 views

Questions about Lebesgue integral of increasing measurable function converges in measure

I'm doing an exercise in Lebesgue integral. Definitions probably used: Some of them can be acquired here: Questions of an exercise in Lebesgue integral. Definition of Convergence in measure: ...
-1
votes
1answer
51 views

Calculating Lebesgue integral 2

How can we calculate Lebesgue integral; for example, when we have $\mu(x)=x^2$ and $ f(x)=x $ $(\int f(x) \mu(dx) = \int x (dx)^2)$ ? can we use $ \int f(x) \mu(dx)=\int fd\mu $ ,instead of ...
0
votes
0answers
22 views

Non-Integrability by the comparaison theorem and contradiction

(H.A.Priestley Introduction to integration 18.4.3) Let $f(x) = \frac{x sin(x)}{1+x^2}$ on $\mathbb R$. We claim that f is not integrable. Note that $f$ behaves like $g(x)=\frac{xsin(x)}{x^2}$ for ...
0
votes
1answer
53 views

counter examples for Riemann and Lebesgue integrabilities

I'm seeking interesting examples that are not mentioned in usual real analysis texts. It seems that in general there is no relationship between Riemann integrability and Lebesgue integrability when ...
0
votes
1answer
24 views

Questions of 2nd exercise in Lebesgue integral

This exercise is the other one that I have some trouble with. Some usual definitions can be acquired here: Questions of an exercise in Lebesgue integral, (obviously, that was the first one which I ...
2
votes
1answer
39 views

Questions of an exercise in Lebesgue integral

I'm doing exercises related to Lebesgue integral and get stuck by two of them. I can't figure out what do some steps in solutions mean. Some definitions probably will be used: Definition of ...
5
votes
3answers
102 views

Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Problem statement: Suppose that $\mu$ is a finite measure. Prove that a measurable, non-negative function $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < ...
1
vote
1answer
57 views

How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?

$f$ is a given function that belongs to $R[a,b]$. For each $n\in\mathbb{N}$, let $P_n$ be the partition of $[a,b]$ $$P_n=\{ a = x_0 < x_1 < \cdots < x_{2^n} = b\}$$ such that $$x_k - ...
0
votes
1answer
129 views

New definition of Lebesgue integral

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
0
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1answer
61 views

How to prove an inequality of Lebesgue integral?

Definition of measurable set: A set $E$ is measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given ...
3
votes
1answer
43 views

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y)dy = 0$ for all $x$. Then $f = 0$ a.e.

This is a corollary to a proof in Bass, but I don't understand why it follows from the proof he gives. I follow everything up until the last statement. Why is it that proving that the integral is $0$ ...
0
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0answers
14 views

A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
0
votes
0answers
25 views

Integral of sup of directed family of elementary functions

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
1
vote
1answer
29 views

What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

$L^2(\mathbb R)$ is isomorphic to $\ell^2(\mathbb R)$ (which has the cardinality of $\mathbb R$ since there is an injection to the space of continuous functions which has the cardinality of $\mathbb ...
2
votes
2answers
70 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...
-1
votes
1answer
61 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [closed]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
4
votes
2answers
79 views

How to show $\int_{[0, +\infty)} \frac{2}{1+x^2} dx$ Lebesgue integrable?

Definition of Lebesgue integral of simple function: We say that a simple function $\psi$ is Lebesgue integrable if the set $\{\psi \ne 0\}$ has finite measure. In this case, we may write the ...
1
vote
1answer
18 views

Continuity of a parametrized surface integral of a sobolev function

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in ...
1
vote
2answers
37 views

Is $f$ integrable if it is the limit of integrable functions with a uniform bound on their integrals?

Let $f_n$ is a sequence of measurable functions on a measure space $(X,\mathcal{B},m)$ converging pointwise to a function $f$. Suppose that $f_n$ is integrable for all $n$ and ...
0
votes
0answers
49 views

What are some functions $f \in L^\infty(\Omega)$

In my text book all it said about $L^\infty(\Omega)$ space is that it is the space of all measurable functions that are bounded almost everywhere. No example given. I can't see how any member of this ...
1
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1answer
28 views

Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
4
votes
3answers
122 views

What does $dx$ mean in a Lebesgue integral?

This is an introduction for Lebesgue integral of simple function in Carothers' Real Analysis. We say that a simple function $\phi$ is Lebesgue integrable if the set {$\phi$ $\ne$ 0} has finite ...
2
votes
1answer
27 views

Can someone present a visualization of the partitioning of a $L^p$ space into equivalent classes?

I am a bit confused by what it means for a $L^p$ space to be partitioned into equivalent classes instead of functions. I understand that give two or more functions $f$, $g$, $h,\ldots$ of which are ...
1
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1answer
54 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
2
votes
2answers
149 views

Can someone show me why mathematicians use $d\mu$ instead of $dx$ for Lebesgue Integral over $u(x)$

I am an engineer and I learned my Lebesgue integral from an engineering text which dumbed down a lot of stuff, most prominently all Lebesgue integrals were introduced as $\int_\Omega u(x) dx$ instead ...
1
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2answers
33 views

Sigma algebra definition and Lebesgue integration

I know the definition of the $\sigma$-algebra, and I have seen it used in integration theory. However, I do not understand why it is defined the way it is. From what I understand, the definition ...
5
votes
2answers
61 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
3
votes
2answers
79 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
0
votes
2answers
29 views

Prove that orthonormalsystem is an orthonormalbasis

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
0
votes
2answers
41 views

Product Integral: Integrability

Given measure spaces $X$ and $Y$. Then it holds: $$\int_Y\int_X|\eta(x,y)|\mathrm{d}\mu(x)\mathrm{d}\nu(y)<\infty\implies\int_X|\eta(x,y)|\mathrm{d}\mu(x)<\infty\quad(y\in Y)$$ Can this ...
1
vote
0answers
33 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
1
vote
3answers
91 views

Lebesgue integration calculation problem?

Let $f:[0,1]\to \Bbb R$ be a bounded, Lebesgue measurable function with satisfies $$\int_{[0,1]}f(x)x^kdx=\frac{1}{(k+2)(k+3)}=\frac{1}{k+2}-\frac{1}{k+3} $$ for each $k\in \Bbb N \cup{0}$. Show ...
1
vote
1answer
69 views

Measure converges to zero

I'm trying solving the following problem: Let $f:[0,1]\to \Bbb{R}$ be a measurable question such that $f(x)>0$ a.e. Let $\{E_k\}_{k=1}^\infty\subset [0,1]$, a sequence of set such that ...