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.

learn more… | top users | synonyms

0
votes
1answer
39 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 ...
1
vote
1answer
51 views

Monotone convergence theorem assuming convergence in measure

I have heard that the monotone convergence theorem holds if the hypothesis of almost everywhere convergence is replaced by convergence in measure. I concur; if $f_n$ converges in measure then there ...
1
vote
1answer
35 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$, E = \bigcup_1^n{E_j}\}$ ...
1
vote
2answers
35 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 ...
0
votes
1answer
31 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 ...
0
votes
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 ...
5
votes
2answers
82 views

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

This is an old preliminary exam problem: Show that, for every nonnegative Lebesgue integrable function $f:[0,1]\rightarrow \mathbb{R}$ and every $\epsilon>0$ there exists a $\delta>0$ such ...
0
votes
1answer
34 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 $f(0)=0$. ...
1
vote
1answer
20 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 ...
3
votes
1answer
38 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 ...
4
votes
2answers
55 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 ...
1
vote
1answer
34 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 ...
1
vote
1answer
29 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 ...
3
votes
1answer
57 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 ...
1
vote
0answers
44 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 $\int_{-\infty}^{\infty} f' (x)dx=0$. ...
0
votes
0answers
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 $g$ ...
0
votes
1answer
35 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 ...
0
votes
1answer
27 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 ...
0
votes
0answers
55 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 ...
2
votes
0answers
59 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
votes
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\}) = ...
2
votes
1answer
43 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 ...
3
votes
1answer
104 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 $$ g(x) = \int_x^a{t^{-1}f(t)dt} $$ ...
2
votes
1answer
29 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 ...
1
vote
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 ...
4
votes
1answer
42 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 ...
1
vote
1answer
43 views

Question on Lebesgue integration and continuity

$(X,\mathcal A, \mu)$ is measurable space. Suppose that function $f(t,x)$ is measurable w.r.t variable $x$ for each $t\in (a,b)$. a) $f(t,x)$ is continuous at $t_0$ for all $x\in X$, and ...
1
vote
3answers
91 views

Lebesgue integration calculation problem?

Let $f:[0,1]\to \Bbb R$ be a bounded, Lebesgue measurable function with satisfies $$\int_{[0,1]}f(x)x^kdx=\frac{1}{(k+2)(k+3)}=\frac{1}{k+2}-\frac{1}{k+3} $$ for each $k\in \Bbb N \cup{0}$. Show ...
1
vote
1answer
24 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 ...
1
vote
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 ...
0
votes
2answers
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 ...
0
votes
0answers
15 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?
3
votes
1answer
64 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 ...
12
votes
1answer
138 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 ...
0
votes
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 $f$ ...
10
votes
1answer
168 views

Significance and applications of the Riesz Representation Theorem in locally compact Hausdorff spaces

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
1
vote
2answers
32 views

Sigma algebra definition and Lebesgue integration

I know the definition of the $\sigma$-algebra, and I have seen it used in integration theory. However, I do not understand why it is defined the way it is. From what I understand, the definition ...
2
votes
0answers
56 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 ...
1
vote
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 ...
1
vote
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 ...
-1
votes
3answers
59 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 $\mathbb R$?
0
votes
1answer
24 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 ...
1
vote
1answer
121 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 ...
0
votes
0answers
27 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 ...
3
votes
2answers
53 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, ...
1
vote
1answer
57 views

How to prove that $\phi_n\uparrow f$, where $f\in R[a,b]$ and $\phi_n$ is like below?

$f$ is a given function that belongs to $R[a,b]$. For each $n\in\mathbb{N}$, let $P_n$ be the partition of $[a,b]$ $$P_n=\{ a = x_0 < x_1 < \cdots < x_{2^n} = b\}$$ such that $$x_k - ...
2
votes
1answer
22 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?
4
votes
2answers
81 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 ...
0
votes
0answers
48 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 $K \subset\Bbb{R^3},$ $$\lim_{n\rightarrow\infty}\int_{K}{|f_n(x)|dx}=0.$$ ...
1
vote
0answers
47 views

Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

For what nonnegative integer $n$ and positive real $c$ does the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral and when does it converge as an ...