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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49 views

Is $\frac{1}{\sqrt x}$ locally integrable?

Can we say since its anti-derivative is like $\frac{1}{2}\sqrt x$, even if we study the measurable bounded set near zero, the anti-derivative will not go to infinity, and hence the original function ...
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1answer
58 views

Almost everywhere continuous functions

A function that is almost everywhere continuous is in $L_2$; however, the converse might not be true. I couldn't find any example to show this, could you help me with this? $$L_2= \left \{ ...
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1answer
25 views

Calculating surface integral (with gauss's theorem (?))

I would like to solve the following problem: Let $B_1$ be the unit ball in $R^3$ and $A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\})$. Let $F(x,y,z) := (-y+e^{x+z}, 0, e^{x+z})$. ...
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1answer
39 views

Are these functions Lebesgue Integrable over these domains?

link to original image $\frac{\sin x^2}{x}$ on $(1,\infty)$; $\frac1x\sin\frac{1}{x^2}$ on $(0,1)$. Hi, I've managed to prove that $\frac{\sin(x)}{x}$ is integrable on $(0,R]$ and is not ...
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2answers
32 views

Calculating 2-dimensional integral

I would like to calculate $\int_A (6yx+2y) d(x,y)$, where $A:= \{(x,y) \in R^2 | x^2/4 + y^2 \leq 1\}$. Can anybody help me with this problem?
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2answers
69 views

Show these functions are/ aren't Lebesgue Integrable? How do I go about showing this?

I have recently been learning the comparison test, MCT, Fatou's Lemma and DCT for Lebesgue integrals, but have been struggling with the details of the proofs. 1) f is 0 a.e. so is integrable to 0 ...
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3answers
65 views

If $f$ is bounded non-negative $L^1$, is $f\leq g$ a.e. for some continuous integrable $g$?

Suppose $f\in L^1(\mathbb{R})\cap L^\infty(\mathbb{R})$ is bounded, non-negative and integrable (w.r.t. Lebesgue measure) : does there exist $g$ continuous (non-negative) and integrable such that $$ ...
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1answer
42 views

If a sequence of functions is zero almost everywhere and converges pointwise almost everywhere, does the same hold for a limit?

Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions in $\mathcal{L}^p(\mathbb{R})$. Each $f_n$ is zero almost everywhere. Additionally, the sequence converges pointwise almost everywhere to some ...
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1answer
35 views

Is the subspace of $L^2([0,1])$ of all “functions” vanishing on $[0,1/2]$ closed?

I am trying to understand the following example from my lecture notes: $\mathcal{H} := L^2([0,1],\lambda)$, then $$K := \{f \in \mathcal{H} \colon f(x) = 0, \text{ for } 0 \leq x \leq ...
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1answer
12 views

Necessity of measurable property

Consider the definition of the Lebesgue integral for a positive function $X\rightarrow [0,+\infty]$: $$ \int f(x) d\mu=\sup_{g\in S, \forall x : g(x)\leq f(x)} \left(\int g(x) d\mu \right)$$ where ...
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1answer
86 views

Proof of regular version of the Urysohn lemma

I know it's a well-known result, but I have not found any clear formalization, and I need a clear formalization. So I want to know if you agree with this formalization, and this proof. Thank you for ...
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0answers
31 views

Theorem 5.3 in Royden 3rd real analysis, pages 100 — 101.

The theorem stats that Let $f$ be an increasing real-valued function on the interval $[a,b]$. Then $f$ is differentiable almost everywhere. The derivative $f'$ is measurable, and $\int^{b}_a f'(x) ...
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19 views

Lower bound of a Lebesgue measure set

Given a sequence of times $T_n\rightarrow \infty$ and $\int_{-T_n}^{T_n} ||u_n(t)||_{L^\infty(R)} dt\geq cT_n$ for some constant $c$. Also, assume that $u(t,x) = O(1)$ for all $t,x\in R$. Prove that ...
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1answer
43 views

prove absolute integrability given square integrability

am trying to follow the outline of a proof in a book i am reading - must be missing something obvious, but would like to understand what exactly... $f$ is complex and square integrable over e. g. [0, ...
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1answer
47 views

Prove that $\int_E fd\mu = \lim \int_E f_n d\mu$ for all measurable set $E$

This is problem 4T in Bartle's The elements of integration and Lebesgue measure. Suppose $f_n$ are non-negative measurable function such that $(f_n)$ converges to $f$, and that $$\int fd\mu = ...
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15 views

Square Integrable Functions Formula

The book reads as follows: "Let $a(x)$ and $b(x)$ be square integrable functions defined on [$a , b$]. First we note that it follows from the elementary inequality $|ab| \le 1/2 (a^2 + b^2)$ ...
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2answers
52 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
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2answers
54 views

In which way Lebesgue Integral integrates over values?

A special tutorial for full dummies says In order to distinguish between the Lebesgue and Reimann integrals consider the values that the function f can take to be on the x-axis (called the ...
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1answer
31 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
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1answer
37 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
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1answer
39 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
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1answer
40 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
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2answers
61 views

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued ...
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0answers
45 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
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0answers
61 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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2answers
37 views

Could fast or irregular oscillations make Lebesgue integral fail?

Let's consider real measurable functions defined in a bounded interval. As long as a function is bounded, oscillations at least cannot make the volume under the graph of the function infinite. But I'm ...
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2answers
44 views

$f$ has a zero integral on every measurable set. Prove $f$ is zero almost everywhere

I am trying to solve the following exercise: Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$]. I have the following proof but it ...
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2answers
86 views

Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$

Find: $$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$$ The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ ...
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3answers
59 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
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1answer
56 views

Counting measure on sigma algebra power set of natural numbers .

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
21 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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1answer
32 views

A limit of integral

How can I prove that $ \lim_{ n\to \infty} \int_{R}^{} \cos(nt)f(t)dt = 0 $ for any $f \in L_{1}(R) $? I believe that I should use a fact that cosinus is a cyclic function and divide this integral ...
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3answers
120 views

Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.

Prove $\displaystyle\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}\,dx$ exists. I want to use Dominated convergence theorem to show the ...
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0answers
21 views

How to integrate with a matrix in the measure?

I've been given the following integral (actually a path integral from quantum field theory). $$ Z(a;N) = \int d^{2n}M.exp(-\frac{1}{2}tr(M^2)-\frac{a}{M}tr(M^4)) $$ where M is a square matrix of size ...
3
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1answer
38 views

Does $\int f(s) ds = \int g(s) ds \not =0$ imply $f(s)=g(s)$?

specifically for an improper integral, but I'm also wondering about for definite integrals. I'd guess that it's true, but I feel like there must exist different functions that integrate to the same ...
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0answers
55 views

Interchange of integral and differentiation: calculus version vs. analysis version

There are theorems that specify conditions that guarantee the interchange of differentiation and integral: $$ \frac{d}{dx}\int f(x, y) dy = \int \frac{\partial}{\partial x}f(x,y) dy $$ In calculus $f$ ...
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2answers
43 views

Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
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0answers
37 views

Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow f$ a.e. [duplicate]

Let $m$ be Lebesgue measure on $\mathbb R$ and let $f_n ,f \in L^1 (m)$. Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow ...
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1answer
32 views

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following $1) x^k f(x) \in L^1([0,1])$ for all $k\in \mathbb{N}$ $2) \lim_{k\rightarrow\infty}\int_{0}^{1}x^k f(x) dx = 0$ $3)$ If ...
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2answers
180 views

Helmholtz theorem

I have been told that the Helmholtz decomposition theorem says that every smooth vector field $\boldsymbol{F}$ [where I am not sure what precise assumptions are needed on $\boldsymbol{F}$] on an ...
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1answer
23 views

Prove $F$ is in $L^1$

This is an old qualifier exam question at my school Let $f \in L^{1}([0,\infty))$ and for $x\geq 0$, define $F(x) = \int_{(x,\infty)} f(t) e^{x-t} dm(t) $ Show that $F \in L^{1}([0,\infty))$ The ...
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0answers
42 views

Fubini's Theorem and expectation of random variables

I have a question regarding the application of the Fubini's Theorem to the expectation of the product of two random variables. Let $X,Y$ be two random variables defined on the probability space ...
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1answer
46 views

Prove that $\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$

Prove that: $$\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$$ I am self-learning these stuff, and I would like to check whether I did things right. Here's ...
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1answer
43 views

$\left\|f\right\|_{L^1(μ_1)}<∞$ $μ_2$-a.e.,$\left\|f\right\|_{L^1(μ_2)}<∞$ $μ_1$-a.e. $⇒$ $\left\|f\right\|_{L^1(μ_1\otimesμ_2)}<∞$

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space and $f:\Omega_1\times\Omega_2\to\mathbb R$ be measurable with respect to $\mathcal A_1\otimes\mathcal A_2$. Can we conclude, that ...
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1answer
35 views

Fubini-Tonelli theorem and absolutely Lebesuge integrable functions

As far as I know, a measurable function is Lebesgue integrable if and only it is absolutely integrable. It is simply because the definition of the integrability requires each of the positive part and ...
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0answers
35 views

Application of Leibniz rule for Lebesgue integral

Consider the real-valued random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the function $f\colon\mathbb{R}\rightarrow [0,\infty)$. Let ...
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0answers
21 views

Different definitions of expectations: which types of integral do they involve?

Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. The probability space induced by $X$ is $(\mathbb{R}, ...
2
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4answers
81 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
0
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2answers
49 views

Why the following integral is Riemann integrable but not Lebesgue integrable?

I know Riemann integrable implies Lebesgue integrable, but why the following integral is Riemann integrable but not Lebesgue integrable? S=$\int_E {1\over{x-y}}dm$, where $E=[0,1]\times[0,1] $. I ...
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0answers
46 views

Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...