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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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 ...
1
vote
1answer
46 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
1answer
21 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
vote
2answers
18 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 ...
2
votes
2answers
99 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 ...
5
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2answers
57 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 $[0, 1]$ ...
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 $L^q$, ...
3
votes
1answer
27 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta\left( ...
3
votes
2answers
77 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
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2answers
28 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
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2answers
38 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
32 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
76 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
2answers
58 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
4
votes
5answers
74 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
6
votes
4answers
2k views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
1
vote
1answer
66 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 ...
2
votes
2answers
41 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwarz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
2
votes
1answer
23 views

Relating Integration by Substitution to Change of Variables Theorem

I'm having trouble relating the change of variables theorem from measure theory to the integration by substitution formula taught in Calculus. I've always thought they were basically saying the same ...
4
votes
2answers
62 views

A problem on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
3
votes
1answer
97 views

Simple Functions: L1-Cauchy?

Given a measure space $(\Omega,\mathcal B(\Omega), \mu)$. Consider simple functions: $$s\in\mathcal{B}(\Omega,\mathbb{C}):\quad s=\sum_ka_k\chi_{A_k}$$ Suppose it is I-Cauchy: ...
9
votes
2answers
2k 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
1answer
48 views

Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
2
votes
2answers
28 views

Restriction of measure arising in Riesz's theorem to Borel sets

Riesz's theorem on representation of positive linear functional on locally compact space as stated in Rudin's "Real and Complex Analysis" assures us that certain $\sigma-$algebra containing all Borel ...
1
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2answers
60 views

Is there a strictly monotone, integrable function $f: \mathbb{R} \rightarrow [0,\infty)$?

Im not sure about the above question. Im guessing that there is none, else the question would probably not be asked that way, but i can't really pinpoint where the contradiction lies.
1
vote
1answer
29 views

$L^p$ and $\ell^p$ spaces

I'm confused. I've read that for $1\leq p<q<\infty$ following inclusions are true: $$\mbox{1)}\qquad \ell^p\subset\ell^q$$ $$\mbox{2)}\qquad L^q\subset L^p$$ My question is - why inclusions ...
6
votes
2answers
110 views

Generalized convex combination over a Banach space

The Question: Is the following true? If not, what further hypotheses do I need? Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, ...
5
votes
2answers
69 views

Prove the following inequality: $\int_{(a,b)}f\ d\lambda\cdot\int_{(a,b)}\frac{1}{f}d\lambda≥(b-a)^2$

Assignment: Let $-\infty < a < b < \infty$ and $f: (a,b) \rightarrow (0,\infty)$ be measurable, such that $f$ and $\frac{1}{f}$ are Lebesgue integrable. Prove the following inequality: ...
1
vote
1answer
20 views

A Change of Variable/Fubini's Theorem

A line in a text reads $$\int_{0}^{\infty} \mu (B(x,u^{\frac{-1}{t}}) du = t\int_{0}^{\infty} r^{-t-1} \mu (B(x,r)) dr.$$ I set $u=r^{-t}$. But then $du=-tr^{-t-1} dr$. Where is the negative?
1
vote
1answer
62 views

Show that $f(x,y) = \frac{e^x+1}{x}$ is Lebesgue integrable.

Show that $$ f: [0,1)^2 \rightarrow \mathbb{R}, \quad f(x,y)=\left\{\begin{array}{cl} \frac{e^x+1}{x}, & \mbox{for }y \le x\\ 0, & \mbox{else} \end{array}\right. $$ is Lebesgue integrable and ...
2
votes
0answers
38 views

minimal distance betwen a point and and the halfspace containing a convex set

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and $$\min_{f\in S} \|f-g^*\|$$ has the unique solution $f^*\in S$. ...
0
votes
0answers
29 views

Condition for $\overline{M}$-measurable in problem 2.24 by Folland

I'm self-learning Real Analysis using Real Analysis of Folland, and I got stuck on this problem. Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X) < \infty$, and let $(X, ...
1
vote
2answers
10 views

Countable additivity with respect to integrands in Lebesgue integrals

The following property of Lebesgue integrals is true for nonnegative measurable functions $f_n$ (because it is a consequence of the monotone convergence theorem): $$\int (\sum_{n=1}^\infty f_n) d\mu ...
6
votes
1answer
181 views

Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ that converges weakly to $u\in W^{1,p}_0(\Omega)$ and converges strongly to $u$ in $L^p(\Omega)$. We define a function $f:\Omega\times ...
0
votes
1answer
11 views

Completion of R-integrable functions by L-integrable functions

I read that: "There's an analogy between the completion of rational numbers by real numbers and the completion of Riemann integrable functions by Lebesgue integrable functions". Can someone elaborate ...
3
votes
1answer
31 views

Prove continuity of averaging function for integrable $f$

I want to prove the following statement which is part of a lemma in my textbook: Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le ...
0
votes
1answer
18 views

Complex Measures: Lebesgue

Given a Borel space $\Omega$. Consider a complex measure: $$\mu:\mathcal{B}(\Omega)\to\mathbb{C}$$ Regard a sequence: $$\eta_n\in\mathcal{L}(\Omega):\quad\eta_n\to\eta$$ Suppose one finds: ...
0
votes
1answer
16 views

Borel Measures: Pushforward

This thread is Q&A. Problem Given Borel spaces $X$ and $Y$. Consider a Borel measure: $$\mu:\mathcal{B}(X)\to\mathbb{C}:\quad\mu\geq0$$ Regard a pushforward: ...
2
votes
1answer
47 views

Complex Measures: Pushforward

Problem Given Borel spaces $X$ and $Y$. Consider a complex measure: $$\mu:\mathcal{B}(X)\to\mathbb{C}$$ Regard a pushforward: $$h\in\mathcal{B}(X,Y):\quad\nu:=\mu\circ h^{-1}$$ Then one has: ...
1
vote
1answer
40 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
1
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1answer
20 views

Bound on the product of functions in $L^1$

Let X be a bounded subset of $\mathbb{R}$ and let $f, g,$ and $h$ be real valued functions in $L^2(X)$. Consider $$\| fgh\|_{L^1(X)}.$$ The hope is to get an upper bound in terms of ...
1
vote
1answer
22 views

Vanishing measure sets and Expectation

During my research, I was required to prove a particular result. I shall just ask what I needed for my result to hold. Let $X_n$ be a sequence of random variables that are integrable and suppose we ...
1
vote
1answer
83 views

Find Limit Using Lebesgue Dominated Convergence

I'm trying to find the following limits using Dominated Convergence Theorem, but can't seem to find a dominating function. Any guidance would be greatly appreciated! $\lim\limits ...
2
votes
0answers
124 views

Question on product measure

Let $(\Omega_1,\Sigma_1,\mu_1)$ and $(\Omega_2,\Sigma_2,\mu_2)$ be two totally finite measure spaces (which implies that $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras). (As usual ...
1
vote
1answer
33 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
5
votes
2answers
66 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 ...
1
vote
1answer
20 views

Support of convolution

Assume $u \in L^1(\mathbb{R}^n)$ and $\mathrm{ess\,supp}(u) \subset U,$ where $U$ is a bounded open set. Now we compute the convolution of $u$ with a function $\eta \in C(\mathbb{R}^n)$ with ...
1
vote
2answers
23 views

What is “an increasing sequence of step functions”?

I'm reading Alan Weir's "Lebesgue Integration and Measure". In exercise 8 on page 30 he talks about "...an increasing sequence of step functions $\{\phi_n\}$..." and "...an increasing sequence of ...
1
vote
1answer
28 views

Questions about a dominated convergence theorem problem

The problem is to find the derivative of Gamma function $\Gamma (y) = \int_0^{ + \infty } {{e^{ - x}}{x^{y - 1}}dx} $ using dominated convergence theorem. Although the following content is lengthy, ...
3
votes
0answers
55 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

Problem Let $f_n\in C[0,1]$. Show that $f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$. Background Let $X$ be a normed space. ...