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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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 ...
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0answers
45 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 ...
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1answer
25 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?
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97 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 {$\phi$ = 0} has finite ...
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1answer
23 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 ...
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1answer
48 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 ...
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2answers
117 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 ...
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19 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 ...
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2answers
58 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 $[0, 1]$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$ ...
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2answers
29 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 ...
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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.
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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 $D$, ...
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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 ...
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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?
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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 ...
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0answers
39 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 $f^*\in S$. ...
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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 ...
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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, ...
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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: ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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0answers
132 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 ...
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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 ...
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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 ...
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2answers
68 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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. ...
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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 ...
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2answers
72 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
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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 $f:\mathbb R\to\mathbb R$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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