0
votes
1answer
34 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
0
votes
0answers
27 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
-1
votes
0answers
25 views

change of variable in the integral with measure $\mu$ [on hold]

Let $f:(X,\mathbb{X}, \mu) \to Y$ a measurable function and $\nu(A)=\mu(f^{-1}(A))$. show that $$\int_Yg \,d\nu = \int_X g(f(x)) \,d\mu(x)$$
0
votes
1answer
22 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
2
votes
0answers
45 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
10
votes
0answers
296 views
+400

$I=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
11
votes
1answer
255 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
1
vote
1answer
44 views

Prove the following sobolev inequality

Let f be a fubction on $[0,1]$ that is continuous and has a continuous derivative f'. show that: $\sup_{0 \leq x,y \leq 1}|f(x)-f(y)|\leq ||f'||_2$. Do not know where to start. Any hint or help is ...
1
vote
1answer
28 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
3
votes
1answer
21 views

Convergence in average on every set implies convergence?

Let's say we're working in a measure space $(X, \mathcal{B}, \mu)$, and let $f_n, f$ be measurable. Suppose I have that, for any measurable set $E$, $$ \int_E f_n d \mu \to \int_E f d \mu $$ Does that ...
3
votes
2answers
62 views

A problem on calculating integral

Show that the integral $$\int_{0}^{1} \frac{1}{x} \left|\cos \frac{1}{x^2}\right|\ dx$$ is finite. I plotted the graph, but it looks like it is infinity.
1
vote
0answers
18 views

Leibniz's Derivative Rule for Integral in Measure Theory

I saw the extension of Leibniz rule for integrals for measure theory on Wiki, although I am not sure if the proposition there is correct. Besides there is no proof for it. Can anybody please introduce ...
1
vote
0answers
12 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
1
vote
0answers
32 views

Measure inequality implies integral inequality?

Let $f$ and $g$ be non-negative, integrable functions on a measure space with measure $\mu$, and suppose there is some constant $c > 0$ such that for every $t \geq 0$, the inequality $\mu(\{f \geq ...
1
vote
1answer
20 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...
1
vote
1answer
37 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
0
votes
1answer
56 views

Integral with respect to a non-standard measure

Let $\mu:P(\mathbb R) \to [0,+\infty]$ be a measure defined by: $$ \mu (\{ \tfrac 1n \})= \tfrac 1n $$ and $\mu(E)=0$ if $E \cap \{ \tfrac 1n \}_{n \in N_0} =\emptyset$ Compute $$\int_{\mathbb ...
1
vote
3answers
48 views

Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
4
votes
1answer
47 views

Show that $g\in\mathcal{L}^q(\mu)$.

Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that $$\int |fg|d\mu\leq C\|f\|_p$$ for ...
3
votes
5answers
49 views

real analysis, chebyshev's inequality

Suppose $f$ is a non negative integrable function on a measure space $(X,M,μ)$. Prove that: $$\lim_{t \rightarrow \infty} t\cdot \mu(\{x:f(x)\geq t\} )=0.$$ Can you help me please?
10
votes
0answers
265 views

Is this $really$ a categorical approach to $integration$?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
2
votes
1answer
77 views

$f(z):=\int_{\mathbb{R}} \frac{1}{t-z} d\mu(t)$ show $\lim_{y\rightarrow 0}iyf(iy)=-\mu(\lbrace 0 \rbrace)$

I have some trouble with part b) Let $\mu$ be a finite Borel measure (i.e finite measure on the $\sigma$-algebra of Borel sets on $\mathbb{R}$). Define the function $$f(z):=\int_{\mathbb{R}} ...
1
vote
0answers
45 views

Integration question measure theory

For the function $$ f(x) = \begin{cases} \infty & \text{if $x=0$} \\ 1/x & \text{if $x \in \mathbb{Q} \smallsetminus 0$} \\ 0 & \text{Otherwise} ...
3
votes
1answer
57 views

Measure Theory & Integration

Let $\Omega$ = {$\omega$$_i$}$^{\infty}_{i=1}$ be a countable set. Let $\mathscr{A}$ := $\mathscr{P}$($\Omega$) be the power set of $\Omega$ Let $\mu$ be a measure on ($\Omega$, $\mathscr{A}$) ...
1
vote
2answers
43 views

Dominated convergence under weaker hypothesis

Let $f_n,\,n\in\mathbb{N}$ be a sequence of real integrable functions, $f_n\to f$ pointwise as $n\to\infty$. The dominated convergence theorem states that if there exists $g\in L^1$ such that ...
1
vote
1answer
36 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
1
vote
0answers
17 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
28 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
0
votes
0answers
28 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
0
votes
1answer
27 views

Scaling the Lebesgue-Stieltjes integral

Suppose that $F$ is a distribution function. Denote by $\mu_F$ the measure on $\mathbb{R}$ induced by $F$. Suppose that $a>0$. Define a new distribution function $F_a$ by $F_a(x):= F(ax)$, and ...
1
vote
1answer
40 views

Signed measure defined by an integral

Let $ (X,\mathcal{M},\mu)$ be a measure space and let $ f:X\to[-\infty,+\infty]$ be an integrable function (i.e. at least one of $ f_+ $ and $ f_-$ is integrable). I want to prove that $ ...
1
vote
0answers
17 views

Prove that $v \bot \mu$ iff $|v| \bot |\mu|$

The title says it all. This problem is quite easy with signed measure, but I can't solve it in complex measure. In complex measure, the only way to get $|v|$ is to express $dv = fd\mu$, then $d|v| = ...
2
votes
0answers
36 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
2
votes
1answer
47 views

Variant of dominated convergence theorem

There are several variants of dominated convergence theorem. The standard one requires $f_n \to f$ a.e. and $|f_n|\leq g$ a.e. where $g$ is integrable. It can be weakened to only convergent in ...
1
vote
1answer
41 views

Using Riemann integral to define Lebesgue Integral

In the text I'm working through, the Lebesgue integral is related to the Riemann integral as follows: For some non-negative, real valued function $f$ on $\Bbb{R}$, set $E_y=\{x:f(x)>y\}$ and ...
1
vote
0answers
30 views

Inverse map measurable

We said that a function $f:X \rightarrow \mathbb{R}$ is measurable iff we have that for all $I_a:=(a,\infty)$, $a \in \mathbb{R}$ $f^{-1}(a,\infty)$ is measurable. Now I want to show that ...
0
votes
0answers
27 views

Showing that two functions defined as limits are equal almost everywhere

Given a bounded function $f: [a, b] \rightarrow ℝ$, let $H(x)=\lim_{\delta \rightarrow 0} \sup_{|y-x| \leq \delta} f(y)$, $h(x)=\lim_{\delta \rightarrow 0} \inf_{|y-x| \leq \delta} f(y)$; $g = ...
1
vote
0answers
30 views

Finite convergence of integral over a space implies convergence of integral over all measurable functions in space.

I'm trying to prove the following: Suppose $\{f_n\}\in L^+$, $f_n \to f$ pointwise, and $\int f = \lim \int f_n < \infty$. Then $\int_Ef = \lim \int_E f_n$ for all $E$ in $\mathcal M$. $L^+$ ...
2
votes
1answer
66 views

Understanding Lebesgue Integration

I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral: In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: ...
1
vote
2answers
55 views

Showing that a function is Lebesgue integrable (or not)

How would I go about showing that the function $ f(x) = \left\{ \begin{array}{lr} &1 & : x \in [n,n+1) & :n\quad \text{even}\\ &-1 & : x \in [n,n+1) & ...
0
votes
0answers
20 views

Summing the area under $x^{-a}$ using rectangles

I'm trying to approximate the area under $x^{-a}, x>1$ as a limiting sum of a finite set of rectangles. If I set the step size of each rectangle as $\alpha$, then the sum of the areas of a finite ...
0
votes
1answer
51 views

Confusion on a integration problem

Is it correct to ask to evaluate the integral: $$\int_{-1}^{1}\dfrac{1}{x}dx$$ The function $f(x)=\dfrac{1}{x}$ is not defined at $0$.
1
vote
0answers
54 views

Fat Cantor-Lebesgue function

I came across the following theorem the other day, "If $f:[a,b]\to \mathbb{R}$ is monotonic increasing, then $f$ is differentiable a.e." If the take the standard Cantor-Lebesgue function then I see ...
1
vote
1answer
26 views

Geometric question involving integral of a function and its inverse.

I am given a function $\phi(s)$, continuous and strictly increasing with $\phi(0) = 0$, and want to show that for all $a,b \geq 0$, $$ab \leq \int_0^a \phi(x)dx + \int_0^b \phi^{-1}(x)dx.$$ I know how ...
0
votes
1answer
25 views

Integral Estimate Using a Function and its Inverse

I want to show the following: given a measure space $(X,\mu)$ and $f,g$ $\mu$-measurable functions on $X$, $$\int_X |f(x)g(x)| d\mu(x) \leq \frac{1}{2}\int_{|f(x)| \leq 1} |f(x)|^2 d\mu(x) + ...
0
votes
1answer
60 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
1
vote
0answers
23 views

How do i prove this stage proving Fubini's theorem for complete measure space?(Edited)

I thought i proved it, but i found an error when i was revising my note. Let $(X,\mathfrak{M},\mu),(Y,\Sigma,\nu)$ be sigma-finite complete measure spaces. Let $\overline{\mu\times\nu}$ be ...
0
votes
0answers
20 views

A condition for a function to be complete measurable

I was doing this exercise in Real Analysis of Folland and got stuck on this problem. I get no clue about how to define these 2 functions series, so I hope some one can help me solve this. I really ...
1
vote
1answer
91 views

Prove that $\int_{E}f =\lim \int_{E}f_{n}$

I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate. Consider a ...
1
vote
3answers
36 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...