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.

learn more… | top users | synonyms

4
votes
1answer
33 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
73 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
27 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
37 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
66 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
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
1answer
22 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 ...
3
votes
1answer
46 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
vote
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.
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 $D$, ...
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 ...
1
vote
1answer
19 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
61 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
37 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 $f^*\in S$. ...
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 ...
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, ...
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: ...
-1
votes
0answers
58 views

Any comment on an integral [closed]

Any solution/approximation for $$ \int_{1}^{\infty} \! r \exp\left\{- \left( a r +b r^{-c} \right) \right\} \, \mathrm{d}r \:,$$ where $a \geq 0$, $b>0$, and $c>2$ are constant?
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
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: ...
1
vote
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
115 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 ...
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 ...
6
votes
2answers
65 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 ...
4
votes
5answers
73 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 ...
4
votes
2answers
61 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 ...
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. ...
3
votes
2answers
39 views

A weaker form of Lebesgue's differentiation theorem in $\Bbb R ^n$

If $f : \Bbb R ^n \to \Bbb C$ is locally-integrable then Lebesgue's differentiation theorem says that $$\lim \limits _{r \to 0} \frac 1 {\lambda \big( B(x, r) \big)} \int \limits _{B(x, r)} f \Bbb d ...
5
votes
2answers
69 views

Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
2
votes
1answer
26 views

A kernel to guarantee integrability

In trying to answer this question, I thought that it might be useful if there exists a function $K:\mathbb R^2\to\mathbb R$ such that for any continuous function $f:\mathbb R\to\mathbb R$, ...
1
vote
2answers
56 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 ...
2
votes
0answers
55 views

If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
1
vote
1answer
54 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
0
votes
0answers
27 views

The infimum of the upper sums of an upper continuous function equals the Lebesgue integral?

Let $f:[a,b] \to \mathbb{R}$ be bounded and upper-continuous. Then it is Lebesgue-integrable. Is the infimum of the upper Darboux sums of $f$ equal to the Lebesgue-integral of $f$? The upper Darboux ...
-3
votes
1answer
33 views

How can I prove that $f$ and $g$ are measurable functions [closed]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
2
votes
2answers
66 views

Prove that $\iint\limits_ {[0,1] \times [0,1]} \frac{x^2-y^2}{(x^2 + y^2)^2}\,\mathrm dx\,\mathrm dy$ is not integrable

I have to prove that the following integral does not exist: $$\iint \limits _{[0,1] \times [0,1]} \frac{x^2-y^2}{(x^2 + y^2)^2}\,\mathrm dx\,\mathrm dy .$$ I think I can use Fubini's Theorem, ie. if ...
5
votes
2answers
74 views

If one side of $\int f\ d\lambda = \int f\ d\mu - \int f\ d\nu$ exists, does the other one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
-1
votes
1answer
51 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$ [closed]

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
1
vote
0answers
19 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
3
votes
1answer
26 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( ...
4
votes
1answer
61 views

$L^{\infty}$ norm

For Lebesgue $p$-integrable functions, what would be the formula for $$\left(\int_0^1 \sum_{i=1}^n | f_i(x)|^p dx\right)^{\frac{1}{p}} $$ as $p\to +\infty$? Would it be $$\max_i \sup_{[0,1]} ...
0
votes
1answer
55 views

Lebesgue differentiation

Some days back I was doing the lebsegue integration. I was amazed by the integration's ability to maximize the potential of Reinman integration. Are there any new differentiation (except the metric ...