1
vote
1answer
25 views

Proving the $u$-substitution formula for Lebesgue integrals

Following the proof at Wikipedia, I'm trying to verify the proof of the substitution rule for integrals under the (fairly simple) assumptions that $\phi:[a,b]\to(c,d)$ is continuous on $[a,b]$ and ...
2
votes
0answers
35 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
2
votes
3answers
84 views

Calculate $\int\frac{dx}{x\sqrt{x^2-2}}$.

The exercise is: Calculate:$$\int\frac{dx}{x\sqrt{x^2-2}}$$ My first approach was: Let $z:=\sqrt{x^2-2}$ then $dx = dz \frac{\sqrt{x^2-2}}{x}$ and $x^2=z^2+2$ $$\int\frac{dx}{x\sqrt{x^2-2}} ...
3
votes
0answers
83 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
0
votes
1answer
42 views

An issue with $\infty \cdot 0$ in showing that Cartesian product of a set with a null set has measure zero

Here is the problem: Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces. Furthermore $A\in \mathcal A$ and $N\in \mathcal B$ such that $\nu(N)=0$. Let ...
0
votes
0answers
36 views

What is this called specifically?

Imagine you take a radius from the center of the shape, you add up all of the lines as it rotates 360 degrees. The radius is measured from its point of rotation, like (0,0) in Cartesian coordinates,to ...
1
vote
0answers
41 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
2
votes
0answers
49 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
1
vote
1answer
25 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
1
vote
1answer
113 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
1
vote
1answer
88 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
1
vote
1answer
47 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
0
votes
1answer
31 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
2
votes
0answers
54 views

$f\in L^1(0,\infty)$ monotone, show $\lim_{x\rightarrow \infty} xf(x) = 0$ [duplicate]

Here is the solution: First $f$ is monotone and integrable on $(0,\infty)$, wolg we can assume that $f>0$ and approaches $0$ as $x$ goes to infinity. Observe that $$xf(2x) \leq \int_x^{2x} ...
0
votes
2answers
35 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
1
vote
1answer
11 views

Clarification about equality regarding integrals

I'm reading Brezis ch. 8 and got stuck in a passage of lemma 8.2 pag. 205. Let $I=(a,b)$, let $g \in L_{loc}^1(I)$, for a fixed $y_0 \in I$, set $$ v(x) = \int_{y_0}^x g(t)dt \ \ \ \ \ \ \ \ \ x \in ...
4
votes
3answers
189 views

Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
1
vote
0answers
36 views

If $f(x)$ is integrable on $[a,b]$ then $c\cdot f(x)$ is also integrable and $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$

I proved the first part of this theorem which says that $c\cdot f(x)$ is integrable,but how to prove that $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$? Maybe it provides a bit help if i tell how ...
4
votes
1answer
77 views

Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
0
votes
0answers
40 views

Area of solid of known cross-section

I was looking at surface areas of solids of known cross-section (that is, you take a region in the xy plane, set up cross sections perpendicular to the region that are defined as a function of x, and ...
3
votes
3answers
58 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
0
votes
1answer
50 views

Proving $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x

Suppose f has a continuous second derivative. Prove $f(x) = f(0) + f'(0)x + \int_0^x (x-t) f''(t) dt$ for all x. Can someone check this for me? What I started with is that if we let $h(x) = \int_0^x ...
1
vote
0answers
27 views

Check my answer - show a function is integrable and find the integral

Let $Q =[0,1]$x$[0,1]$. Let $f: Q \to \mathbb R$ defined as such: if $(x,y) \in \mathbb Q$x$\mathbb Q$ then $f(x,y)=\frac{1}{n_1}+\frac{1}{n_2}$ where $x=\frac{m_1}{n_1}$ and $y=\frac{m_2}{n_2}$ are ...
1
vote
1answer
50 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
1
vote
0answers
28 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
2
votes
1answer
114 views

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
1
vote
0answers
25 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...
0
votes
1answer
54 views

Prove the integral of $\cot x$ is $-\ln|\csc x|+C$

I know how to prove it to be $\ln|\sin x|+C$, but I do not know the method to prove it this way. thanks
1
vote
1answer
61 views

how do you show one definite integral is less than another definite integral over the same region?

How do you show $$ \int ^{b}_{a} f(x) \,dx \leq \int ^{b}_{a} g(x) \,dx$$ if you know f(x) and g(x) are continuous over [a,b] and $$f(x) \leq g(x)$$ for $${a \leq x \leq b}$$ Here is the way I ...
0
votes
0answers
58 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
0
votes
2answers
544 views

Prove that $\int_0^{2\pi} \cos \theta e^{\cos\theta} \cos(\sin \theta) - \sin \theta e^{\cos\theta} \sin(\sin \theta)\,\mathrm{d}\theta$ equals zero.

Note this question is a part of a bigger question, I have done a) and b). Problem 2 Let $u(x,y) = x e^x \cos y - y e^x \sin y $ a) Show that u is harmonic in the entire plane b) Find a ...
1
vote
1answer
33 views

Domination $\Rightarrow$ $0$ equality

Let $\phi \in C^{\infty}_c(\mathbb{R})$. In class, my teacher said that the dominated convergence theorem (DOM) may be used to prove that $$ \lim_{\epsilon \to 0^+} \int_{-\epsilon}^{\epsilon} \! \log ...
1
vote
1answer
125 views

Seeking domination?

Define $f_n : \mathbb{R} \to \mathbb{R}$ by $$ f_n(x) := \int_{-n}^{nx} \! \frac{\sin t}{t} \, \mathrm{d}t. $$ I'm searching for a function $g \in L^1_{\text{loc}}(\mathbb{R})$ (that is, a real ...
5
votes
2answers
77 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...
3
votes
2answers
97 views

Proof of integral

Is there an analytical method to show that $$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$ for $a>0$. I have confirmed this result numerically for a range of values of $a$. ...
3
votes
1answer
37 views

Integrability of time differences via bootstraping?

the question is somehow inspired by the Alt-Luckhaus paper (Lemma 1.5) Let $B:\mathbb{R}\to\mathbb{R}$ be continuos and nonnegative, $\Omega\subset \mathbb{R}^n$ a bounded domain, $h,T>0$. Let ...
2
votes
1answer
48 views

Error in theorem 3-8 “Calculus on manifolds”

This error is also commented in the Addenda of the book, with a proposed solution, but I don't understand how it should fix the proof. The (part of the proof of the) theorem involved is: Suppose ...
1
vote
0answers
44 views

Non-negative, continuous function with integral [duplicate]

Let there be an integrable, non-negative function $f$ in a range $[a,b]$. If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous. I have ...
0
votes
0answers
70 views

Box-sum criterion related problem

Define a function $f$ by $$f(x) = \begin{cases}42 & \text{if }x =1,2,3,4; \\0 & \text{otherwise} \end{cases}$$ Prove that $f$ is integrable on $[0,5]$ by using the box-sum criterion. My ...
2
votes
0answers
66 views

Proving the converse of the Cauchy criterion for integration

Prove the converse of the Cauchy criterion for integration. That is, prove that if $f$ is integrable on $[a,b]$, then for any $\epsilon>0$ there is a $\delta>0$ so that for any partitions ...
0
votes
1answer
77 views

Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$

I'm reading Real Analysis by Royden 4th Edition. The entire problem statement is: Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ ...
1
vote
0answers
115 views

$\int f^2(x)dx=0$ if $\int f(x) x^n=0 $ for all $n=1,2,3,\dots$ (TIFR GS ($2012$))

Question is to check if for a continuous function $f: [0,1]\rightarrow \mathbb{R}$ $$\int_0^1 f^2(x)dx=0 \text { given that }\int_0^1 f(x) x^n=0 ~\forall ~n=0,1,2,3,\dots$$ I think it is true and ...
0
votes
2answers
83 views

Show that $\int_{-x}^xf=2\int_0^xf$ implies that $f$ is even.

Please tell me whether the following proof works: $f$ is continuous on $\mathbb R\to\mathbb R$ such that $\int_{-x}^xf=2\int_0^xf~\forall~x\in\mathbb R.$ Show that $f$ is an even function. ...
1
vote
1answer
125 views

Show that the fundamental solution $E_2$ of Laplace equation is local integrable

For $x\in\mathbb{R}^n\setminus\left\{0\right\}$ the function $$ E_n(x):=\begin{cases}\frac{1}{2\pi}\ln\left(\frac{1}{\lVert x\rVert}\right), & \text{ for ...
1
vote
0answers
98 views

Let $f:[a,b]\to\mathbb R$ be bounded and for all $c\in(a,b),~f$ is Riemann integrable on $[c,b].$ Prove that $f$ is Riemann integrable on $[a,b].$

Please tell me whether my proof for the following problem will work: Let $f:[a,b]\to\mathbb R$ be bounded and for all $c\in(a,b),~f$ is Riemann integrable on $[c,b].$ Prove that $f$ is Riemann ...
8
votes
1answer
199 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of ...
3
votes
2answers
91 views

What am I doing wrong?

I am trying to prove the integral test for series, but got a strange result. Assume that $f$ is decreasing and positive. Because the series can be imagined as the area-sum of $1$-wide rectangles of ...
2
votes
1answer
56 views

Proof that the Riemann-Integral satisfies $\int_A \lambda f = \lambda \int_A f$

Suppose $A\subset\mathbb{R}^n$ is a closed rectangle and $f:A\to \mathbb{R}$ is Riemann-Integrable on $A$. I want to show that $\lambda f$ is integrable and that $$\int_A \lambda f =\lambda\int_Af $$ ...
3
votes
4answers
138 views

Show that $\int_{1}^\infty \dotsb\int_{1}^\infty \frac{dx_1 \dotsb dx_n}{x_1^{\alpha_1}+\dotsb + x_n^{\alpha_n}}<\infty$

Here's my solution to an old qualifier problem. Would you tackle it differently? Is there a flaw in my work? Suppose that $\alpha_1, \dotsc, \alpha_n$ are positive numbers such that ...