Tagged Questions
1
vote
1answer
56 views
Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = ...
2
votes
1answer
53 views
Simplification of an expression
How do I simplify the following expression?
$$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$
where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e ...
0
votes
1answer
34 views
Show that E is measurable?
Suppose $E_1= [1, 1 \frac12] , E_2 = (2, 2\frac14), E_3 = [3, 3\frac18], E_4 = (4 , 4 \frac{1}{16}) , \dots , E= \bigcup_{n=1}^{\infty}E_n
$
i) Show $E$ is measurable
ii) Compute $m(E)$
Here is ...
2
votes
1answer
51 views
Riemann integral with intervals?
Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$
Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$.
You can use the definition of Riemann integral ...
0
votes
0answers
45 views
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
Show that
$\int_{0}^{1} dx \int_{0}^{1} f(x,y) dy=\frac{\pi}{4}$
$\int_{0}^{1} dy \int_{0}^{1} f(x,y) dx=-\frac{\pi}{4}$
2
votes
1answer
51 views
Interchange differential operator with Lebesgue integral.
Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
0
votes
1answer
39 views
Absolute Convergence of a Function
I have got stuck with a question. Please help me.
Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$.
Thank You.
0
votes
0answers
54 views
Change of differentation and integration signs.
I'm going through an old exam in a course I'm taking. I have the given rule:
Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
2
votes
0answers
63 views
Prove Heisenberg uncertainty principle (measure and integration theory)
Here is a question in measure and integration theory,
Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
0
votes
1answer
68 views
Whether convergence in L2 norm implies convergence a.e.? [duplicate]
How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
1
vote
1answer
48 views
If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.
Please help me with this problem!
Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$.
If ...
1
vote
1answer
58 views
Meaure theory problem no. 12 page 92, book by stein and shakarchi [duplicate]
Show that there are $f \in L^1(\Bbb{R}^d,m)$ and a sequence $\{f_n\}$ with $f_n \in L^1(\Bbb{R}^d,m)$ such that $\|f_n - f\|_{L^1} \to 0$, but $f_n(x) \to f(x)$ for no $x$.
1
vote
1answer
41 views
Lebesgue integrable function and limit
Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$.
My solution which is ...
2
votes
1answer
42 views
Continuity of $\max$ of Lebesgue integral
Let $m$ be a probability measure on $Z \subseteq \mathbb{R}^p$, so that $m(Z)=1$.
Consider a locally bounded $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, with $X \subseteq \mathbb{R}^n$, ...
3
votes
2answers
79 views
If $f$ and $g$ are integrable, then $h(x,y)=f(x)g(y)$ is integrable with respect to product measure.
I have to show that $h$ is measurable as well as $\int h d(\mu \times \nu) < \infty$ .
I tried showing by contradiction that $\int h$ had to be finite but I'm stuck with showing how it is ...
3
votes
1answer
35 views
Sum of measures and integral
Suppose we have a measure that can be expressed as linear combination of 2 measures: $m=am_1+bm_2$. What does this imply for the calculation of the integral? Do we have:
$$\int f\,dm=a\int f\,dm_1 + ...
3
votes
2answers
63 views
Extracting a subsequence from a sequence of $\mathcal{L}^1$ functions
Any help with the following problem is appreciated.
Given: a sequence of nonnegative functions $(g_n)$ which are U.I. (uniformly integrable) in $\mathcal{L}^1(0,1)$ with $\sup_n \Vert g_n \Vert_1 ...
2
votes
1answer
92 views
Lebesgue Convergence Theorem
I need some clarifications about the Lebesgue Convergence theorems, I am using the book Beginning Functional Analysis by Karen Saxe and I believe that she has stated the monotone convergence theorem ...
1
vote
1answer
40 views
$L_p$ spaces and convergence
The Riesz-Fischer Theorem implies that Lp-convergence implies pointwise
a.e. convergence of a subsequence.
There is an example that shows that the converse may not be true...
Let E = [0, 1], $1 ...
1
vote
1answer
56 views
Characteristic Function in Product Measure
Let $X=Y$ be the interval $[0, 1]$, with $A=B$ the class of Borel
sets. Let $\mu$ be the Lebesgue measure and $v=c$ the counting
measure. Show that the diagonal $\Delta = \{(x,y) | x=y\}$ is
...
1
vote
1answer
74 views
Measurable with respect to counting measure
$\Bbb N$ is the set of natural numbers. Let $(X, A, \mu) = (Y, B, \nu) =
(\Bbb N, M, c)$ be the measure space such that $M = 2^N$, and $c$ the
counting measure defined by setting $c(E)$ equal ...
3
votes
2answers
114 views
Is $\int fg$ equal to the supremum of $\int f_0g_0$, over simple functions $f_0\leq f, g_0\leq g$?
Let $f, g$ be positive functions on a measure space $X$ with measure $\mu$. I have seen the definition that
$$\int f \,d\mu = \sup\limits_{0 \leq h \leq f} \int h \,d\mu$$
where the supremum is over ...
0
votes
1answer
21 views
Outer measure defined by a continuous and bijective function
This problem is from K.T. Smith's Primer of Modern Analysis:
Let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be continuous and one-to-one on an open set $\Omega \subset \mathbb{R}^d$ and define $$\nu(A) ...
1
vote
1answer
59 views
Lebesgue Integral on Lebesgue measurable set satisfies Caratheodory condition
Let $f$ be a non-negative, measurable, and integrable over every compact set in $\Omega$, where $\Omega$ is an open set $\subset \mathbb{R}^d$.
For every Lebesgue measurable set $E$ (abbreviated as ...
1
vote
4answers
48 views
Bounded and finite integral
If $f:\mathbb R \rightarrow [0,\infty]$ is bounded and $\int f<\infty$, then $\int f^2<\infty$.
I'm trying to write $f$ in terms of a finite linear combination of characteristic functions, ...
2
votes
1answer
108 views
Riemann-Stieltjes integrability criterion
I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7:
Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
3
votes
1answer
75 views
Using a complete measure space, show the existence of sequences of functions
Fron Folland's Real Analysis
Let $(X,\mathcal{M},\mathcal{u})$ be a measure space with $\mathcal{u}(X) < \infty$, and let $(X,\mathcal{\overline{M}},\mathcal{\overline{u}})$ be its completion.
...
0
votes
1answer
39 views
Question regarding application of Tonelli Theorem
Hi I have a question below, I am wondering if anyone would help me with it, thank you in advance!
Prove that for any independent random variable x, y then
$$\int_{\mathbb{R}}F_{x} dP_{y} = ...
2
votes
0answers
64 views
Riemann Stieltjes integral definition and implications
I am studying the Riemann Stieltjes on Tom Apostol's book mathematical analysis second edition and I have a the following question.
Given $[a,b]$ we define a partition of this interval to be a set $P ...
1
vote
1answer
113 views
$f_n$ converges pointwise to $f$ implies integral $f_n$ converges to integral $f$
Let $\lambda(E)< \infty$ and $f_n$ be measurable and continuous (on $E$) for each $n\in\mathbb{N}$.
If $f_n$ converges pointwise to $f$ (continuous on $E$) for all $x\in E$, then $\int_Ef_n ...
0
votes
1answer
24 views
Fourier series inequality with polynomial
I have the following question:
Let f be in $\mathbf{L}_{\mathbf{R}}^2([-\pi;\pi])$. Show that
$$\left({\int_{-\pi}^\pi |x^nf(x)|\,\mathrm{d}\lambda(x)} \right ) \leq \frac{2*\pi^{2n+1}}{2n+1} ...
1
vote
2answers
156 views
Question 2.1 of Bartle's Elements of Integration
The problem 2.1 of Bartle's Elements of Integration says:
Give an exemple of a function $f$ on $X$ to $\mathbb{R}$ which is not
$\boldsymbol{X}$-mensurable, but is such that the function $|f|$ ...
1
vote
0answers
83 views
Characterization of Dirac Measure
Let $x_0$ be a point in a set $X$ and $\delta_{x_0}$ the Dirac measure concentrated at $x_0$. Characterize the nonnegative real-valued functions on $X$ that are integrable over $X$ with respect to ...
1
vote
2answers
69 views
Graph of a measurable function
I am reading through Terence Tao, and I was wondering how one would prove that if $f:\mathbb{R}^d \rightarrow [0, \infty]$ is measurable, then the area under $f$ is a measurable subset of ...
1
vote
1answer
45 views
$A\subset[0,1]$ with measure $<1$ so that $\int_A f(x) dx = \int_{[0,1]} f(x) dx$ for continuous $f$?
Is there a set $A\subset[0,1]$ of measure $<1$ so that $\int_A f(x) dx = \int_{[0,1]} f(x) dx$ for continuous $f:[0,1] \to \mathbb{R}$?
$A^c$ must have empty interior for sure. (Otherwise support ...
3
votes
1answer
114 views
Change of variables for a Dirac delta function
I have often seen the following equality in Physics textbooks.
$$\int_{\mathbb{R}}\delta\left(\alpha x\right)f\left(\alpha x\right)|\alpha|dx=\int_{\mathbb{R}}\delta(u)f(u)du$$ or ...
1
vote
0answers
77 views
0
votes
1answer
82 views
Derivative inside an integral
Assume that I have an integral
$$
I=\int_\Omega f(\omega)g(\omega)d\omega,
$$
where $\Omega$ is a measure space and $\omega\in \Omega$.
What is
$$
\frac{\partial I}{\partial f(\omega)}?
$$
i.e. I ...
0
votes
1answer
66 views
Integral to get the derivative of the gamma function
To prove that the gamma function is differentiable (through differentiation under the integral) I need to show that for all $\beta \in (0,1)$ the function
$\displaystyle ...
6
votes
1answer
126 views
Integration theory
Any help with this problem is appreciated.
Given the $f$ is measurable and finite a.e. on $[0,1]$. Then prove the following statements
$$ \int_E f = 0 \text{ for all measurable $E \subset [0,1]$ ...
1
vote
1answer
39 views
Prove the limit of the integral
I need to prove that
$\lim_{n\rightarrow \infty} \int_0^{n^2} n\sin(x/n) e^{-x^2}dx=1/2$
I tried substituting $t=\frac{x}{n^2}$, but it was not very useful because I can't find a bounding function ...
1
vote
3answers
85 views
Continuity of the lebesgue integral
How does one show that the function, $g(t) = \int \chi_{A+t} f $ is continuous, given that $A$ is measurable, $f$ is integrable and $A+t = \{x+t: x \in A\}$.
Any help would be appreciated, thanks
0
votes
0answers
37 views
Integration in euclidean space as the average along lines
Let $f(z_1,\ldots, z_n)$ be a polynomial with complex coefficients. Let $d G_n=\exp(-|z|^2) d V_n$ be the Gaussian measure on ${\mathbb{C}}^n$ where $d V_n$ is the Lebesgue measure. Let $\Omega$ be ...
3
votes
0answers
104 views
An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.
We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
4
votes
1answer
55 views
If $\mu$ is a signed finite measure, then $\|\mu\| = \sup \left\{ \int f d\mu : |f| \leq 1 \right\} $
If $\mu$ is a signed finite measure, then
$$
\|\mu\| = \sup \left\{ \int f d\mu : |f| \leq 1 \right\}
$$
I did the inequality "$\geq$". Somebody help with the other inequality "$\leq$"?
0
votes
1answer
49 views
Bounding the integral of $\exp(-x^2)$
Let $a \in \mathbb{R}$ and $r>0$. Could we find a constant $C>0$ and $0<n<1$ such that \begin{equation}\int_{a-r}^{a+r} e^{-x^2} dx \leq Cr^n.\end{equation}
for all $r>0$.
My effort: I ...
2
votes
1answer
124 views
Characteristic function of the Smith-Volterra-Cantor set
Let the characteristic function of the SVC set be denoted by $ \beta $. Does the Riemann integral $ \displaystyle \int_{0}^{1} \beta ~ d{x} $ exist? I think it does since $ \beta $ is bounded, but I ...
0
votes
1answer
56 views
The relation between arbitrary measure space and the Lebesgue integral
Let $(X, \mathcal F, \mu)$ be a measure space and $f\in M^+(X,\mu)$ (the measurable non-negative functions), and $t>0$. Now let $$S_f(t)=\{x\in X:f(x)>t\} \quad \Psi_f(t)=\mu(S_f(t))$$
Prove ...
0
votes
1answer
93 views
Application of Dominated Convergence Theorem.
Find $L=\lim\limits_{n \to \infty} \int_0^{n a} \exp\left(-\dfrac{t}{1+\frac{b t}{n}}\right) dt$, where $a>0$, $b>0$.
I can't see what is dominating function, but I feel that I have to use ...
0
votes
1answer
37 views
On the convergence of a specific sequence of integrable functions
Let $\{f_n\}$ a sequence of measurable non-negative functions on $\mathbb{R}$ converging point-wise on $\mathbb{R}$ to $f$, and let $f$ integrable over $\mathbb{R}$. If $\displaystyle ...
