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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12 views

Convergence of a subsequence of a subsequence of distribution functions

I'm trying to find a solution for the following problem: Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of signed (Baire)-measures (of bounded variation) on $[a,b]$ and let $F_{\mu_n}(t):=\mu_n([a,t))$ ...
3
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1answer
42 views

Lebesgue integral of a strange function.

The problem statement is as follows: Let $f: [0, 2]\to \mathbb R_{+}$ be defined by $f(t)=m(\{x\in [0, \pi]: t\leq 1+\cos (3x)\leq 3t\}).$ Compute $\int_0^2 f(t)\,dt$. I'm not certain how to begin ...
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0answers
12 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
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1answer
11 views

Obtaining Essential Range and Support of a Measurable Function from Estimate

The following is an old real analysis qual problem which I cannot solve. Problem. Let $f\geq 0$ be a measurable function on $\mathbb{R}^{n}$. Suppose there exists $C>0$ such that for all ...
4
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1answer
43 views

integrability of $f^3$ for some Lebesgue measurable function

I'm trying to solve the following problem from an old qualifying exam, but nothing I've tried has been successful, so any help would be greatly appreciated. Suppose $f: \mathbb{R} \rightarrow (0, ...
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1answer
21 views

Set of positive measures and Banach space

In measure theory i heard recently a statement in my class, which says that the set of all (positive) measures does not make a Banach space ( whereas the set of signed measures makes up a Banach space ...
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1answer
23 views

Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuos function on [a,b], let $\mu$ be a bounded Borel measure on [a,b], and let $\Phi_\mu(t)=\mu([a,t))$ with ...
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0answers
28 views

Help With Limit of Integral

So I am working through some practice problems, and on one of them I can't get the second part: For $x\in(0,\infty)$ and $n\in\{1,2,3,\dots\},$ let $f_n(x)=\frac{e^{\sin\left({x^2/n}\right)}}{1+x}.$ ...
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1answer
22 views

Problems in the integration limits to apply Fubini's theorem

If $f:(0,a)\rightarrow\mathbb{R}$ integrable function and $$g(x)=\int_{x}^a \dfrac{f(t)}{t}dt.$$ Then $g$ is integrable and $\int_{0}^a g(t)dt=\int_{0}^a f(t)dt$. I have to use Fubini's theorem but ...
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0answers
8 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
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0answers
27 views

Continuous convergence [on hold]

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
18 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 ...
2
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1answer
42 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
16 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
30 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
28 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
34 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
64 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
405 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 ...
5
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2answers
45 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 ...
2
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1answer
37 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
36 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*} ...
1
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1answer
41 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 ...
0
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3answers
30 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 ...
2
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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 ...
3
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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 $\int_Kx \,dx \geq \int^{m(K)}_0 x\,dx$, ...
3
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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
14 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
39 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 ...
3
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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
18 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
33 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 ...
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3answers
65 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...
3
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1answer
55 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
53 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
26 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 ...
2
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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 ...
1
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1answer
27 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
107 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 ...
0
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2answers
27 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
43 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} ...