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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41
votes
2answers
16k 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 $L^q$, $\lim_{p\to\infty}\|f\|_p=\|f\|_\...
50
votes
2answers
16k 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 ...
12
votes
2answers
4k 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? ...
3
votes
2answers
507 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 ...
2
votes
2answers
206 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 ...
2
votes
3answers
188 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
3answers
623 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 = \int^{\infty}_{0}pt^{...
42
votes
8answers
10k 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 ...
4
votes
1answer
172 views

A differentiation under the integral sign

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function Lebesgue summable on all $\mu$-measurable and bounded subsets of $\mathbb{R}^n$, where $\mu$ is the usual Lebesgue measure defined on $\mathbb{R}^n$, ...
5
votes
2answers
2k 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 $L^...
1
vote
1answer
221 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 ...
6
votes
1answer
174 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 $|\...
16
votes
1answer
324 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 ...
10
votes
2answers
5k 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 ...
2
votes
1answer
128 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: $$A_N\uparrow\...
10
votes
1answer
1k views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
7
votes
2answers
2k views

Integral vanishes on all intervals implies the function is a.e. zero

I am having trouble with the following problem: $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$: $$\int_{[0,a]}f\,dm=0.$$ Prove that $f=0$ for $m$ almost every $x$ ...
7
votes
2answers
4k 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 ...
3
votes
1answer
102 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 $F(-\infty) = 0$, $F(\...
1
vote
1answer
139 views

Help evaluating $\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

$$\int_{-1/2}^{1/2}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)df$$ I've tried simplifying the integrand, but I can't get to a point where I can evaluate the integral. I know $\prod_{i=0}^...
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 \...
5
votes
1answer
557 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 ...
23
votes
2answers
723 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 ...
8
votes
2answers
602 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: $$\{...
2
votes
2answers
4k 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 ...
1
vote
2answers
639 views

Integrability of Derivative of a Continuous Function

Let $f$ be continuous on $[a,b]$ and has finite derivative a.e. on $[a,b]$. Let $f_n(x)=n[f(x+1/n)-f(x)] $ s.t. $f_n$ be uniformly integrable on $[a,b]$. I want to show : $f'$ is Lebesgue integrable. ...
7
votes
3answers
1k views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
5
votes
3answers
153 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
5
votes
3answers
2k views

A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$. However Royden & Fitzpatrick’s book "Real Analysis" (4th ...
10
votes
1answer
189 views

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n \...
8
votes
1answer
830 views

Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
5
votes
3answers
182 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 $\...
4
votes
2answers
89 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 $f\...
4
votes
1answer
109 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, $$\lim_{n\...
3
votes
2answers
150 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x \mapsto\int_x^1\frac{1}{t}f(...
2
votes
0answers
259 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
723 views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and $\...
1
vote
2answers
195 views

Helmholtz theorem

I have been told that the Helmholtz decomposition theorem says that every smooth vector field $\boldsymbol{F}$ [where I am not sure what precise assumptions are needed on $\boldsymbol{F}$] on an ...
5
votes
1answer
338 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
83 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
120 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
178 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 ...
3
votes
2answers
234 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 $u\...
3
votes
3answers
147 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 $L^p$ ...
2
votes
3answers
133 views

Translating an integrable function creates a sequence that converges to $0$ almost everywhere

Let $a_n$ be a sequence and $N\in\mathbb{N}$ be fixed. Assume that no more than $N$ of the $a_n$'s lie in any interval of lenght $1$. Show that for any $f\in L^1(\mathbb{R})$, we have $\lim_nf(x+a_n)=...
2
votes
2answers
965 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function f(...
1
vote
1answer
162 views

$C_c^{\infty}(\mathbb R^n)$ 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 ...
8
votes
1answer
1k views

How to find a measurable but not integrable function or a positive integrable function?

For an arbitrary interval $I$, how can we find a positive on $I$ integrable function? And how does one construct a measurable but not integrable function. If not all measurable functions are ...
5
votes
1answer
224 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
2answers
118 views

A question about Measurable function

Let $f$ be a real-valued Lebesgue measurable function on $\mathbb{R}$. Prove that there exist Borel measurable functions $g$ and $h$ such that $g(x)=h(x)$ almost everywhere and $g(x)\le f(x) \le h(x)$ ...