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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45 views

Suppose $f$ is continuously differentiable on $[0,1]$, $|f'(x)|$ is bounded, show that $|\int_0^1 f(x) - \sum^n_{i=1}f(i/n)\cdot 1/n| \le M/n$.

Problem statement: Suppose $f$ is continuously differentiable on $[0,1]$, and that $\sup_{x \in [0,1]}|f'(x)|\le M< \infty$. Show that $$\big|\int_0^1 f(x) - \sum^n_{i=1}f(i/n)\cdot 1/n\big| \le ...
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1answer
54 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
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1answer
44 views

L'Hôpital with absolute continuity

I have been studying for my real analysis qualifying exam, and I have noticed a trend of questions similar to the following: Suppose that $f$ is absolutely continuous, $f'\in L^3$, and $f(0)=0$. ...
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1answer
43 views

Average integral for continuous functions with compact support

Let $f$ be a continuous function with compact support in $\mathbb{R}^n$. Show that \begin{equation} \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)\,dy = f(x), \end{equation} where $B_r(x)$ is the ...
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0answers
53 views

$f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. [duplicate]

Problem: Suppose $f: \Bbb R \rightarrow \Bbb R$ is absolutely continuous on every interval $[a,b]$, and that both $f$ and $f'$ are in $L^1 (\Bbb R)$. Prove that $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
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0answers
25 views

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure and $ f_n(x)=\dfrac{|\cos(x^{-2})|}{x^{1-1/n}} $ for $ x\in (0,1] $ Is there lebesgue integrable function $g$ ...
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1answer
81 views

Example of a function in $L^2(\mathbb{R})$ with derivative not in $L^2(\mathbb{R})$.

We know examples of a function which doesn't lie in $L^2(\mathbb{R})$ with derivatives in $L^2(\mathbb{R})$: $$f_1(x) = \mathrm{arctg}(x) \notin L^2(\mathbb{R}), \qquad f_1'(x) = \frac{1}{x^2+1}\in ...
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1answer
32 views

Can this probability be shown by using the properties of Lebesgue integration

(Grimmett and Stirzaker - Probability and Random Processes - Exercise 1.3.5) I am studying Lebesgue integration in parallel to probability theory, and my question is: Can the following be shown by ...
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1answer
74 views

A problem from Real Analysis of Folland

I got stuck on this problem. For the first statement, I tried to use $\epsilon -\delta$ condition, but still couldn't come to conclusion. So can anyone please help me solve this or give me some clue ...
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1answer
55 views

Showing continuous functions are dense in $L^p([0,1],\lambda)$, where $\lambda$ is Lebesgue measure

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
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2answers
71 views

If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t., (a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded. (b) The sequence ...
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1answer
89 views

There is some known deficiency through the Lebesgue Integral?

The Integral in the Riemann sense has a lot of deficiencies, and the Lebesgue Integral can solve almost all of them. I know that over limited intervals, Lebesgue Integral is a generalization of the ...
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1answer
95 views

Question about notion $d\mu = fdv$ in Real Analysis of Folland

I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and ...
2
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1answer
48 views

Integral should not be continuous.

I'm looking for a counter-example: Let $f:[0,1]\times \mathbb R\to\mathbb R$ be continuous in such a way that $$F(x):=\int_{\mathbb R} f(x,t) dt$$ defines a function $F:[0,1]\to\mathbb R$ (so in ...
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1answer
43 views

Showing a certain function vanishes almost everywhere

Can someone give me a hint on the following problem? I'm not sure what to do... Suppose $f\in L^1([0,1])$ is such that for all $n=0,1,2,...$ we have $$\int_0^1 f(x)(\sin x)^n\,dx = 0.$$ Show that ...
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1answer
39 views

Help verify a solution showing $f\left(x \right)=\int_\Bbb{R} {{\chi _A}\left(y \right){\chi _B}\left( {x-y} \right)dy} $ is well-defined everywhere

The question is, Let $A,B⊂[0,1]$ be measurable sets with $|A|>1/2$,$|B|>1/2$ where $|*|$ denotes Lebesgue measure. Prove that a. $|A⋂(1-B)|>0$ where $1-B≔{1-x:x∈B}$ and conclude that ...
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1answer
60 views

if $\mu(X)$ is finite and $f$ is finite on X a.e then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. Prove that if function $f$ is measurable and finite on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ I have been ...
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1answer
81 views

Mean value theorem for Lebesgue integral

Let $f$ be a mesurable function and $g$ be integrable function, and $\alpha, \beta$ are real numbers such that $\alpha \leq f \leq \beta$ a.e . Prove that there exists a real number $\gamma \in ...
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1answer
111 views

Prove that $\int_0^{a}{\int_x^a{t^{-1}f(t)dt}} = \int_0^a{f(x)dx}$ [duplicate]

I got stuck on this problem from Real Analysis by Folland. Can anybody give me any hints on how to solve this? If $f$ is Lebesgue integrable on $(0, a)$ and $$ g(x) = \int_x^a{t^{-1}f(t)dt} $$ ...
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2answers
41 views

if $\mu(X)$ is finite then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. a) Prove that if function $f$ is measurable on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ b) Can we ...
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0answers
18 views

proof coordinate functions of integrable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ integrable

If $$f(x)=f_1(x_1)\cdots f_n(x_n)$$ and $f$ is an integrable function from $\mathbb{R}^n$ to $\mathbb{R}$. Proof that $f_i(x_i)$, $i = 1, \ldots, n$ are integrable. With the Fubini theorem?
2
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1answer
85 views

Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., show $\mu(\{0\}) = 1$.

Problem statement: Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., whenever $f$ is real-valued, bounded, and integrable. Show $\mu(\{0\}) = ...
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1answer
35 views

How to define a “line” and “symmetry w.r.t. a line” in $L_2(\lambda)$ space

For any $x,y\in L_2([0,1],\lambda)$, define the inner product $\langle. , . \rangle$ by \begin{equation} \langle x, y \rangle=\int_{[0,1]} x(t) y(t) \lambda (dt) \end{equation} Is it proper to ...
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1answer
51 views

prove that $\lim_{m \rightarrow \infty} \Sigma_{k=-m^2}^{m^2}|\int^{(k+1)/m}_{k/m}f(x)dx|=\|f\|_{L^1 (\Bbb R)}$.

Suppose $f \in L^1 (\Bbb R)$, prove that $$\lim_{m \rightarrow \infty} \sum_{k=-m^2}^{m^2}\left|\int^{(k+1)/m}_{k/m}f(x)\,dx\right|=\|f\|_{L^1 (\Bbb R)}.$$ For this one, it's easy to prove when $f$ ...
12
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1answer
156 views

$L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$

I am currently studying for an analysis qualifying exam, and this problem has been bothering me: Suppose we have a sequence $\{f_n\}$ in $L^2(\mathbb{R})$ such that ...
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1answer
52 views

Conditions to ensure nice integrability of supremum of difference on neighborhood

Let $u\in L^1(\mathbb{R}^n)$. Based on that alone, can I say anything nice about the following integral? $$ \int\limits_{\mathbb{R}^n}{\sup\limits_{|y|\le h}{|u(x+y)-u(x)|} \text{ d}x} $$ Ideally, the ...
3
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1answer
67 views

Multiplication operators on $L^2$

Let $X$ be a $\sigma$-finite measure space, and let $g$ a measurable complex-valued function $X$, which lies in $L^\infty(X)$. I would like to determine sufficient and necessary properties for the ...
3
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0answers
63 views

Calculate $\lim\limits_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx$

The problem is to find an integrable function that bounds $f_n(x) = \frac{n^p x^r \log x}{1 + n^2 x^2}$ where $r>0$, $p<\min \{2,1+r\}$ so we can calculate $$\lim_{n \to \infty} \int_0^1 ...
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2answers
69 views

Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$

Problem statement: Find the derivative of $$f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy$$ and find an ordinary differential equation that $f$ solves. Find the solution to this ordinary ...
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0answers
23 views

How is the line integration defined in the most general setting?

A natural generalization of Riemann-Stieltjes integration is Lebesgue integration. Some would say that the use of Lebesgue integration would be overkill when treating differentiable or continuous ...
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3answers
65 views

Lebesgue integrability of $x^{-\frac{1}{2}}$

Is the function $x^{-\frac{1}{2}}$ is Lebesgue integrable over $\mathbb R$ or on some subset of $\mathbb R$?
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1answer
30 views

Prove that the minimal set of functions for which these properties hold are step functions

(H. Priestley - Introduction to Integration - Exercise 4.3) Define the class $\mathbb L$ of integrable functions for which the following $Basic Properties$ hold: (1) Building Block: $ \forall a,b ...
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1answer
47 views

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Prove that $\mu(\Omega)^2\leq\int f \,d\mu\int\frac{1}{f}\,d\mu$. I don't know if that what I did is correct or if it will help to solve the problem, but here it is: Using the Hölder inequality ...
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0answers
57 views

General step functions for lebesgue integral

For simplicity, I will only assume we are talking about the lebesgue integral on the same line. I read a construction of the riemann integral, that was designed in a way to resemble the construction ...
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2answers
57 views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, ...
2
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1answer
28 views

Lebesgue Integral is Linear

I found the following statement without proving or explanation. It says: "the Lebesgue integral is linear." What does it mean? Is it something to prove? If yes, how could we prove it?
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2answers
109 views

If $\sup_n\int_E f_n(x)\ \mathsf dx\leq M\mu(E)$ then the measure of $\{x\in [0,\infty)\mid f(x)>M\}=0$.

This question came up when I was studying for an analysis qualifying exam: Suppose $f_n\geq 0$ for all $n\geq 1$, $f_n\rightarrow f$ a.e. on $[0,\infty)$ and there exists $M>0$ such that ...
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1answer
130 views

Help with a proof that $\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 0$

The question is Let f be a continuous Lebesgue integrable function on $[0,+∞)$, show $\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 0$. My attempt: Suppose $\mathop {\lim ...
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0answers
51 views

If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

I know, if $f\in L$ (the set of all Lebesgue integrable functions), then there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$ and ...
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2answers
53 views

Lebesgue Integral definition In $\mathbb{R}^n$

I am studying Lebesgue Integration in $\mathbb{R}^n$, reading chapters from different books, but still not sure what is Lebesgue integral! Is there an explicit definition for Lebesgue integral, or an ...
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2answers
49 views

A bounded Lebesgue measurable function and integral. [duplicate]

I honestly have no idea where to even begin with this question: Let $f: [0,1]\rightarrow \mathbb{R}$ be bounded and Lebesgue measurable. Suppose that for every $0\leq a<b\leq 1$, we have ...
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0answers
49 views

A question about Lebesgue Integrals

Suppose that $\{f_n\}$ is a sequence of measurable real valued functions on $\Bbb{R^3}$ and satisfies i) For any compact set $K \subset\Bbb{R^3},$ $$\lim_{n\rightarrow\infty}\int_{K}{|f_n(x)|dx}=0.$$ ...
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1answer
14 views

Exponential and convergence in $L^2$ bis

This question is a continuation of my question "Exponential and convergence in $L^2$" posted above: Let $(f_k)$ be a sequence of elements of $L^\infty(\Omega)$, which converge in $L^2(\Omega)$ to ...
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1answer
52 views

I am seeking a solution using Lebesgue Differentiation Theorem to prove that $|S|=0$, and $|\cdot|$ means Lebesgue measure.

Let $S$ be a measurable subset of $\Bbb{R^2}$. Assume for every $x\in S$ there exists a sequence of cubes $\{Q_k(x)\}$ centered at $x$ with side lengths tending to zero such that $$|S\cap Q_k(x) ...
2
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2answers
192 views

The Fundamental Theorem of Calculus (for the lebesgue integral)

I am interested in the following statement. Let $f:[a,b]\rightarrow \mathbb{R}$ be continuous and differentiable. Then, if $f':(a,b)\rightarrow \mathbb{R}$ is integrable, then $$ f(x)-f(a)=\int ...
3
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1answer
38 views

Sufficient conditions for the Hardy-Littlewood Maximal function $M(f)$ being continuous

There are four common versions of Hardy-Littlewood Maximal operator $M(f)$: centered/uncentered + ball/cube. I noticed that the continuity of $M(f)$ depends on the version. For example, let $f$ be the ...
1
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1answer
53 views

Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Let $n \geqslant 1$, Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$, where $u \in (0,1)$. What I did is just use the definition of ...
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1answer
32 views

Is this theorem an extension of Scheffé Lemma

Let $\nu_n(A)=\int_A f_n d\mu$ and $\nu(A)=\int_A f d\mu$ where $f_n, f$ are density functions. If $\nu_n(X)=\nu(X)$ and $f_n\to f$ ($\mu$- a.e) then $$\sup\limits_{A\in \mathcal ...
1
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1answer
69 views

Show that the Fubini Tonelli theorem does not work for this function

Problem Statement: Let $X = Y = R$ and let $B$ be the Borel $\sigma$-algebra. Define $$ f(x,y) = \left\{ {\begin{array}{*{20}{c}} 1&{x \ge 0{\text{ and }}x \le y < x + 1}\\ { - 1,}&{x \ge ...
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1answer
31 views

If sum of a set of increasing functions defined on $[a,b]$ is convergent, will this sum be Lebesgue measurable?

If $\{f_n\}$ is a set of increasing functions defined on $[a,b] \subset \mathbb R$ and $\sum_{n=1}^{+\infty} f_n$ converges to $F(x)$, will $\sum_{n=1}^{+\infty} f_n$ be Lebesgue measurable? I got ...