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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When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
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1answer
46 views

Show $F: (0, \infty) \rightarrow \mathbb{R}$ is diff'ble

Hi just need a bit of help with a few parts of this practice question: Show $F: (0, \infty) \rightarrow \mathbb{R}$ diff'ble with respect to $t \in (0, \infty)$, where $$F(t) := ...
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25 views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but… (see below)?

Is there a sequence $(f_n)$ of Riemann integrable functions such that $\lim f_n(x) = f(x)$ almost everywhere on $[a,b]$ and $\lim\int_a^bf_n$ does not exists in Riemann sense, but it does in Lebesgue ...
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35 views

Show that for every $\epsilon > 0 $ there exists $h \in \mathcal{L}^1(X)$ non-negative and $\delta > 0$ such that:

I am working through some practice questions, and I think I have gotten the first two parts, but I am having trouble deriving the third part: Let $(X,\mathcal{A},\mu)$ be a finite measure space. ...
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72 views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

For which values $\alpha \in \mathbb{R}$ is $f$ integrable? $$f: \mathbb{R}^2 \rightarrow \mathbb{R} : f(x, y) = x \frac{\ln(1 + x^2 + y^2)}{(x^2 + y^2)^\alpha} $$ if $ (x,y) \neq (0, 0) $ and $ ...
3
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1answer
20 views

Can this be proved using the MCT instead of the DCT?

I've seen various version of the DCT prove that if $f$ is a real valued, or extended real valued, or complex, integrable function, and if $\{E_n\}_n$ is a sequence of disjoint measurable subsets, ...
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21 views

Cauchy formula for repeated Lebesgue integration

Is there an equivalent of the Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for the following \begin{equation} f^{(-n)}(x) = \int_a^x ...
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32 views

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$.

Counterexample of the failure of integration by parts when we only assume differentiability of $f$ and $g$. We know when $f$ and $g$ are both AC functions, the integration by parts is true. Is it ...
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21 views

$\mu$ is a finite Borel measure on $\Bbb R$, absolutely continuous w.r.t. to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous.

Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq ...
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45 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
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85 views

Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important?

The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ ...
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35 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
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1answer
13 views

Fractions in L^\infty

Let $f,g \in L^2(\Omega)$, where $\Omega$ is an open bounded subset of $R^n,\; n\ge 1$. We consider the two following properties: (1) $\frac{f}{g} {\bf 1}_{\{x\in \Omega;\; g(x)\ne 0\}} \in ...
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32 views

Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$ where the $f_n$ are measurable functions? ...
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1answer
8 views

Lebesgue-integrability of piecewise function with random variable

This function is Lebesgue-integrable:$$\chi(x)= \left\{ \begin{array}{ll} 1 & \text{if}~x~\text{is rational}\\ 0 & \text{if}~x~\text{is irrational}. \end{array} ...
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15 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
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1answer
40 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
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1answer
41 views

Differentiation Commute with Lebesgue Integration

My question is simple: Given $f: \mathbb{R}^{n+m} \to \mathbb{R}$, $f\in C^{k}(\mathbb{R}^{n+m})$ , and $X \subset \mathbb{R}^{n}$. Write $f$ as $f(x_1, \ldots, x_n, t_1, \ldots t_m)$. When is ...
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39 views

Question about formula for total variation of complex measure from Real Analysis of Folland

Let $\nu$ be a complex measure on $(X, \mathcal{M})$. If $E \in \mathcal{M}$, define: $\mu_1(E) = \sup\{\sum_1^n{|v(E_j)|}:n \in N, E_1, ..., E_n$ disjoint$, E = \bigcup_1^n{E_j}\}$ ...
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38 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
32 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
36 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
21 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
46 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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24 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
58 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
29 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
42 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
33 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

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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60 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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59 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
79 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 ...
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1answer
44 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
30 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
35 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
43 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
37 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
106 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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1answer
33 views

If $f, g$ are measurable functions, then $f+g$ is measurable

Show that $f(x)+g(x)<a$ iff there exists rational number $r,q$ such that $r+q<a$ and $f(x)<r; g(x)<q$. Use this to prove if $f, g$ are measurable functions, then $f+g$ is ...
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31 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
16 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?
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1answer
71 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
30 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 ...
0
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1answer
21 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
139 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
34 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
65 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 ...
2
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0answers
57 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
65 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
20 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 ...