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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Continuous convergence

If f_n converge pointwise to $0$ in $\mathbb{R}^d$, $\int f_n dm =1$ for every $n\in \mathbb{N}$ and $g \in L^1_m \cap C(\mathbb{R}^d,\mathbb{R})$. Then how can I prove that: \begin{equation} \int ...
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1answer
16 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
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1answer
35 views

Example for the benefit from monotone convergence

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann ...
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1answer
15 views

how to construct a monotonic function on a closed interval which is discontinuous at each end points [on hold]

How to construct a monotonic function on a [0,1] which is discontinuous at each end points?
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1answer
29 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
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0answers
27 views

Can I approximate a measurable set with an open set for integration purposes?

I have a Lebesgue measurable function $f:X\rightarrow \mathbb{R}$ where $X\subset\mathbb{R}$. Is there an open set $X^O$ such that \begin{equation*} \int_X f=\int_{X^O} f \end{equation*} and $X^O$ is ...
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1answer
31 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
3
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1answer
63 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
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2answers
398 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
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0answers
9 views

$L_2((-2,2))$ function that has $L_1((-1,1))$ discrete derivative but not derivative

I am trying to find an example of a function $u\in L_2((-2,2))$ such that $||\delta_h(u)||_{L_1((-1,1))}$ is uniformly bounded in $0<|h|<1/2$ but $u'$ is not in $L_1((-1,1))$. Where ...
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2answers
44 views

dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$. Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for ...
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1answer
33 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
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1answer
34 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
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1answer
40 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
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3answers
29 views

Help with Real Analysis Integral

I'm working through practice problems and I came along the following: Evaluate $\lim_{n\rightarrow\infty}\int_0^n(1-\frac{x}{n})^n dx.$ I think this should work out via Dominated Convergence ...
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2answers
47 views

Antiderivative is continuous

The following comes from Bass' book on Real Analysis: (Here $dy$ is Lebesgue measure) Exercise 7.6 Suppose $f:\mathbb{R}\to\mathbb{R}$ is integrable, $a\in \mathbb{R}$, and we define ...
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1answer
40 views

What does “$\mathbb{1}$” mean in this document?

I understand everything in this document on the first page except the following: \begin{align} \overline{\int_a^b}f&=\inf\left\{\int_a^b\psi:\psi\geqslant ...
4
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1answer
31 views

integral over a subset of interval in $\mathbb{R}$

Consider a finite interval $[0,d]$, where $d$ is a positive real number. Let $K$ be a measurable subset of $[0,d]$ Then, how can I prove or disprove that $\int_Kx \,dx \geq \int^{m(K)}_0 x\,dx$, ...
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2answers
37 views

$\overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R)$

While reading a proof in a book they used the following result: $$ \overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R) $$ saying that it's well known !! But all I can see is ...
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0answers
34 views

Fourier transform and $L^1,$ $L^2$ convergence

Let $\phi \in L^2(\mathbb{R})$ and $\hat{\phi}$ be the Fourier transform of $\phi.$ Does this mean that $\sum_{m \in \mathbb{Z}} |\hat{\phi}(x + 2 \pi m)|^2$ converges in the $L^1$ sense on each ...
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0answers
12 views

Interpretation of infinitesimal measure in Lebesgue integration

I have a little trouble understanding the notation of the infinitesimal measure in Lebesgue integration. For example, let's assume I want to compute an volume integral of a function $f: D \rightarrow ...
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0answers
26 views

Reference request: Measure theory books using $\omega(\alpha) = |\{f>\alpha\}|$

I am working from Wheeden and Zygmund's Measure and Integral, and they prove theorems such as $\int_E f = -\int_{-\infty}^{+\infty} \alpha d\omega(\alpha)$ where $\omega(a) = |\{x: f(x)>\alpha\}|$ ...
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0answers
13 views

Dominated convergence theorem in case of converge in measure. [duplicate]

I have heard that the dominated convergence theorem hold if almost everywhere convergence is replaced by convergence in measure. I concur if fn converges to f in measure then there exists a ...
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1answer
38 views

monotone convergence theorem( converges in measure)

I have heard that the monotone convergence theorem hold if almost everywhere convergence is replaced by convergence in measure. I concur if fn converge in measure then there exists a subsequence ...
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1answer
34 views

How to use the Bounded convergence Lemma or the Monotone convergence Theorem to calcuate those Lebesgue Integrals?

$\lim\limits_{n\rightarrow\infty}\int\limits_0^\infty n\sin(\frac{x}{n})(x(1+x^2))^{-1}dx$ $\lim\limits_{n\rightarrow\infty}\int\limits_0^1 \frac{1+nx^2}{(1+x^2)^n}dx$ I have tried to show that the ...
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2answers
39 views

Lebesgue integrable function, convergent series

I am trying to solve the following: Let $(X,\Sigma, \mu)$ be a measurable space, $f:X \to \mathbb R$ measurable and let $A\in \Sigma$. For each $n$ natural number, we define $A_n=\{x \in A: |f(x)| ...
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0answers
16 views

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$,

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$, $\forall g \in L^2({\sigma})$ here $x\in \Sigma$ $\Sigma$ ...
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2answers
32 views

Calculate limit of Lebesgue integrals

I am trying to calculate this limit: $$\lim_n \int_0^{n^2}e^{-x^2}\sin(\frac{x}{n})dx$$ Since $$\int_0^{n^2}e^{-x^2}\sin(\frac{x}{n})dx=\int_{[0,\infty)}e^{-x^2}\sin(\frac{x}{n})\mathcal ...
4
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3answers
63 views

Assume $f(x)\in L^1(0,1)$, prove that $g(x) = \frac{1}{x}\int_{0}^{x}\frac{f(t)}{log(t)}dt$ is in $L^1(0,1)$

Assume $f(x)\in L^1(0,1)$, prove that $g(x) = \frac{1}{x}\int_{0}^{x}\frac{f(t)}{log(t)}dt$ is in $L^1(0,1)$ Have a hard time knowing where to start...
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1answer
54 views

Lebesgue integrable function $g$ equals characteristic function

I am trying to solve this problem: Let $g:[0,1] \to \mathbb R$ be a non negative integrable function over $[0,1]$. Prove that if there is $\alpha \in \mathbb R$ such that for all $n \in \mathbb N$, ...
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0answers
42 views

Lebesgue integration. If $f:(a,b)\to\mathbb{R}$ has a primitive then is locally integrable

This is a problem proposed in the context of Lebesgue integration theory. If $f:(a,b)\to\mathbb{R}$ has a primitive function then it is locally integrable. I only need the case when f is positive ...
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3answers
52 views

Proving Lebesgue Integral Equality

I would love some help with this problem: Let $(X,\mathcal F,\mu)$ be a measurable space and let $f:X\to[0,\infty)$ be a positive Lebesgue integrable function. Prove that $$\int_X fd\mu = ...
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1answer
24 views

if $|f_n|<g \in L^1$, and $f_n \rightarrow f$ in measure, how do we know $\lim_{n\to \infty} \int f_n = \int f$

I know that a subsequence converges, but I am not even convinced that $\int f_n$ converges at all. They are all finite, but I am not certain how to bound them. I have considered working with $\int ...
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1answer
25 views

A counterexample for Lebesgue's Dominant Convergence theorem - where is my mistake?

I am having some trouble with Lebesgue's Dominant Convergence theorem. It seems as if I have a counterexample, and I can't find my mistake. Say that $\mu$ is a uniform measure over ...
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5answers
81 views

Dominated Convergence Theorem

Give an example of a sequence $\{f_n\}_{n=1}^\infty$ of integrable functions on $\mathbb{R}$ such that $f_n \to f$ but $\int f_n \not\to \int f$. Explain why your example does not conflict with the ...
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1answer
26 views

Measure-theoretic analogue of a result from elementary calculus

I recall from elementary calculus being taught about defining the "average" of a continuous function from a compact subset $K = [a, b]$ of $\mathbb{R}$ to $\mathbb{R}$ by $\frac{\int_{a}^{b} f(x) ...
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2answers
106 views

Prove that the Lebesgue integral of $f\chi$ equals to $0$ indicates that $f=0$ a.e.

Suppose $f: [0,1] \to \mathbb{R}$ is bounded, measurable, and $\int_{[0,1]}f \chi_{[0,a)}\, d\mu = 0$ for all $a \in [0,1]$. Prove that $f=0$ a.e. I know that if $\int_{[0,1]}f\, d\mu = 0$, then ...
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2answers
26 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
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0answers
20 views

Heuristic: Daniell integral vs. Lebesgue integral

What are the advantages of the Daniel Integral over the Lebesgue integral and visa-versa? Heuristically speaking, I was wondering why this axiomatic operator is less popular besides the fact that it ...
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1answer
42 views

Evaluating an integral by dominated convergence theorem [closed]

I would like to know how to solve this two problems: a) $$ \lim_{n\to \infty}\int_0^n \left( 1-\frac{x}{n} \right)^{-n}\log{(2+\cos(x/n))} \, dx $$ b) $$ \lim_{n\to \infty}\int_0^{\infty} n e^{-nx} ...
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3answers
37 views

Showing that a sequence of Lebesgue Integrable functions converges to 0 a.e

Let $f_n: E \to \mathbb{R} \cup \{\infty\}$ be Leb.-integrable and suppose: 1) There is a sequence $\{a_n\}$ s.th. $a_n \ge 0$ 2) $\sum_{n=1}^{\infty} a_n = L$ (i.e.: it converges to some L) ...
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1answer
50 views

Computing Lebesgue Integral

I am trying to show that \begin{equation} \int_0^{\infty} \frac{x}{e^x-1} dx = \sum_{n=0}^{\infty} \frac{1}{n^2} \end{equation} Please note that this is the Lebesgue integral. My current strategy ...
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2answers
56 views

Show that the following function is Lebesgue integrable.

\begin{equation} \int_0^{\infty} \frac{x}{e^x-1} dx \end{equation} I know that this function has been tackled from other perspectives, but I haven't been able to find anything on its Lebesgue ...
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1answer
22 views

Calculate the Lebesgue integral of a step function

I'm having some trouble with this problem. Let $$f(x)= \begin{cases} 1 &\text{for}\,\, x = \frac{1}{n}\,,\, n=1,2,\cdots \\ 2 &\text{otherwise} \end{cases}$$ Compute the value of the ...
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1answer
20 views

Why (and how) to choose $a$ in $\varrho(t):=a 1_{[0,1]}(t)\exp\left(\frac 1{t^2-1}\right)$ such that $\int_{\mathbb{R}^n}\varrho(|x|)\;dx=1$?

Let $$\varrho(t):=\begin{cases}\alpha\exp\left(\frac 1{t^2-1}\right)&\text{, if }t\in [0,1]\\ 0&\text{, otherwise}\end{cases}$$ Why (and how) can we choose $\alpha\in\mathbb{R}$ such that ...
3
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3answers
55 views

Sequence of Lebesgue integrals

I am trying to solve this problem: Let $f \in L^1([0,1])$ be a non negative, finite function. Show that $$\lim_{n \to \infty} \int_0^1 \sqrt[n]{f(x)}dx=m(\{x \in [0,1]/f(x)>0\}$$ This is what I ...
3
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1answer
45 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
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1answer
42 views

$Var(F)|_{a}^{b}=\int_{a}^{b}|f|d\alpha.$

Let $[a,b]$ be an interval in $\mathbb R$,and $\alpha :[a,b]\to \mathbb R$ be monotone increasing. Let $f:[a,b]\to \mathbb R$ be integrable, bounded and with respect to $\alpha$. Define $F:[a,b]\to ...
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1answer
52 views

Real Analysis and Lebesgue Measure: Step Functions

For each integer $n$ and $x\in (0,1)$, let $x = 0.k_1k_2k_3...k_nk_{n+1}...$, where $k_i$ is an element of $\{0,...,9\}$ be the decimal expansion of $x$. For such $x$, define $f_n(x) = ...
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1answer
38 views

Limit of sequence of Lebesgue integrals

I am trying to prove the following: Show that for each $g \in L^1([0,\infty))$,$$\lim_{n \to \infty}\dfrac{1}{n} \int_0^n xg(x)dx=0$$ What I did up to now is: Let $h(x)=xg(x)$, then $|h| \leq n|g|$ ...