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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30
votes
2answers
9k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
34
votes
2answers
10k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
10
votes
2answers
3k views

Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
31
votes
9answers
7k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
3
votes
2answers
296 views

Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
0
votes
1answer
126 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
4
votes
2answers
74 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
2
votes
1answer
95 views

$f \in L^1$, but $f \not\in L^p$ for all $p > 1$

"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$." I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
1
vote
3answers
451 views

$L^p$-norm of a non-negative measurable function

Can I ask a homework question here? Let $f$ be measurable and non-negative in $\mathbb R^d.$ Using Fubini's theorem, show that for $1 \leq p \lt \infty,$ $$\lVert f\rVert^p_p = ...
3
votes
2answers
755 views

pointwise convergence and boundedness in norm imply weak convergence

I am contemplating over the following exercise (in which $E=[0,1]$): Let $f_n$ be a sequence of functions in $L^p(E)$, $1<p<\infty$, which converge almost everywhere to a function $f$ in ...
1
vote
3answers
132 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
1
vote
1answer
286 views

Riemann-Lebesgue lemma

How can I prove the following result? Let $([-1,1],M,m)$ a measure space, where $m$ is the Lebesgue measure in $[-1,1]$. If $f$ is Lebesgue integrable, then ...
10
votes
3answers
2k views

Can a function that has uncountable many points of discontinuity be integrable?

First of all, I would like to show you how we defined Riemann-integrals and Lebesgue-integrals to make sure that we are talking about the same: Riemann-intregrability Let $f:\mathbb{R} \rightarrow ...
6
votes
2answers
3k views

Is Dirichlet function Riemann integrable?

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded ...
1
vote
1answer
2k views

Lebesgue Integral but not a Riemann integral

Is it possible for a function to be a Lebesgue integral, but not a Riemann integral? After the comments below I realize my question was not a good one. Thank you. This is my edited version: Let $f$ ...
8
votes
2answers
567 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
6
votes
1answer
162 views

Showing that a function is in $L^1$

I need to prove the following statement or find a counter-example: Let $u\in L^1\cap C^2$ with $u''\in L^1$. Then $u'\in L^1$. Unfortunately, I have no idea how to prove or disprove it, since the ...
2
votes
2answers
3k views

Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$

I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
20
votes
2answers
533 views

Category Theory and Lebesgue Integration.

I'm wondering if there's any Category Theory floating around in the theory of Lebesgue Integration. To avoid things becoming too broad, let's keep this focused on the basics. Here's how I see the ...
13
votes
1answer
221 views

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

Let $f$ be a Lebesgue integrable function on $[0,2]$. If $\int_E fdx=0$ for all measurable set $E$, such that $m(E)=\pi/2$. Is $f=0$ a.e. Prove or disprove I could not figure out anything. Can a ...
9
votes
1answer
174 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} ...
5
votes
2answers
1k views

Riemann-integrable (improperly) but not Lebesgue-integrable

Show that for any $\alpha>1$ the function $$ g_{\alpha}(x):=\sin(x^{\alpha}), x\geq 0 $$ is improperly Riemann-integrable but not Lebesgue-integrable on $\mathbb{R}_+$. Could ...
5
votes
2answers
3k views

Does Riemann integrable imply Lebesgue integrable?

Suppose a definite integral exists in the Riemann sense. Does that mean the integral exists as a Lebesgue integral, and do we get the same result either way? ------- BTW: I have a MS in Electrical ...
6
votes
3answers
171 views

a characterization of $L^p$ space

The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of ...
5
votes
1answer
298 views

Yet another definition of Lebesgue integral

Let $[a, b]$ be a finite interval of the real line. A partition $P$ of $[a, b]$ is a finite sequence of numbers of the form $a = t_0 < t_1 <\cdots < t_{k-1} < t_k = b$ Let $(X, \mu)$ be ...
4
votes
1answer
70 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
4
votes
1answer
118 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
3
votes
1answer
64 views

Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$

Let $f \in L^1(\mathbb{R}^d)$. Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$. What I want to do is bound $|f(x)-f(x-t)|$ above by something and then use the Lebesgue Dominated ...
3
votes
1answer
144 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
1
vote
0answers
96 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 ...
1
vote
1answer
67 views

A continuous $L^1$ function $f : \mathbb{R}\rightarrow \mathbb{R}$..

For a continuous function $f :\mathbb{R}\rightarrow \mathbb{R}$ satisfying $$\int_{\mathbb{R}}|f(x)|dx<\infty$$ and for some $\alpha >0$ let $d_f(\alpha)$ be tthe lebesgue measure of the set ...
5
votes
1answer
376 views

Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...
4
votes
1answer
200 views

Properties of special rectangle (measure)

Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent: a) $\lambda(I)=0$ b) $I^{\circ}=\emptyset$ (i.e., ...
4
votes
1answer
414 views

Extension of Fatou's lemma

let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$. I want to show if (1) holds, then (2) holds too. $$\lim_{n \rightarrow ...
3
votes
3answers
110 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 $L^p$ ...
3
votes
2answers
195 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
3
votes
1answer
98 views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Problem: Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure. My Attempt: Let ...
3
votes
1answer
87 views

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

in preparation for an exam I wanted to show that for $\mu(\Omega)=1$ and $h:\Omega \rightarrow [0,\infty]$ measurable the following inequality holds: $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq ...
2
votes
1answer
65 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
2
votes
3answers
289 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
2
votes
0answers
221 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
votes
2answers
162 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
2
votes
3answers
108 views

summable square function implies…?

I have difficulty to demonstrate this: $$ \int_{-\infty}^{\infty}|f(x)|^2dx<\infty~~~\text{(summable square function)}$$ then, $$\lim_{|x|\rightarrow\infty }f(x)=0$$ thank you.
2
votes
1answer
288 views

Bounded measurable function and integral with charcteristic function

I have been struggling with the following for quite some time now. If anyone can give me some help, it will be much appreciated: Let $f$ bounded, measurable and $E$ be a set of finite measure. Let $A ...
2
votes
3answers
880 views

Must $f$ be measurable if each $f^{-1}(c)$ is?

Suppose $f$ is a real-valued function on $\mathbb R$ such that $f^{−1}(c)$ is measurable for each number $c$. Is $f$ necessarily measurable?
1
vote
3answers
57 views

Let $f \in L^1$ then prove $\lim_{b \rightarrow \infty} \int_b^{\infty} f(x) dx=0$.

So the question is as stated in the title. We are given the hint to use LDCT. Since this is homework I'm not looking for an explicit solution. I just need hints. For example, my first thoughts were ...
1
vote
1answer
83 views

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Let $F \subset \mathbb{R}$ be a closed set and define the distance from $x \in \mathbb{R}$ to $F$ by $d(x,F)= \inf_{y \in F} |x−y|.$ Prove that $$\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$$ for a.e. ...
1
vote
1answer
101 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
1answer
187 views

Prove that $f(x)$ is integrable on $\mathbb{R}$.

Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.
0
votes
0answers
90 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...