Tagged Questions
0
votes
2answers
35 views
Riemann integral and Lebesgue integral
$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that
$$
\int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt
$$
where $m$ is Lebesgue measure.
I know the question may be a little dump.
...
1
vote
1answer
62 views
Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.
Okay, so the question is the following:
Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that
\begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 ...
1
vote
0answers
57 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
vote
1answer
25 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
0
votes
1answer
33 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 ...
0
votes
1answer
56 views
E measurable with m(E) < $\infty$?
Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$.
ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable.
I told my ...
0
votes
1answer
78 views
If $f :\mathbb{R}\to\mathbb{R}$ is measurable, then $E = \{x: f(x) \geq 3\}$ is measurable
Prove: Suppose $f : \mathbb{R}\to\mathbb{R}$ where $f$ is measurable and $E = \{x: f(x) \geq 3\}$. Show $E$ is measurable.
I saw this statement while reading in a paper and thought this might ...
2
votes
0answers
26 views
Lebesgue Integral Rudin Problem [duplicate]
Suppose {$n_k$} is an increasing sequence of positive integers and E is the set of all x$\in$($-\pi, \pi$) at which {sin$n_k x$} converges. Prove that $m(E)=0$.
Hint: For every A $\subset$ E, ...
0
votes
0answers
45 views
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
Show that
$\int_{0}^{1} dx \int_{0}^{1} f(x,y) dy=\frac{\pi}{4}$
$\int_{0}^{1} dy \int_{0}^{1} f(x,y) dx=-\frac{\pi}{4}$
1
vote
0answers
29 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
0
votes
1answer
28 views
How to show this dual space pairing is measurable
If $f \in L^2(0,T;H^{-1})$ and $g \in L^2(0,T;H^1)$, how to show that $\langle f(t), g(t) \rangle_{H^{-1},H^1}$ is measurable over [0,T]?
If it's measurable, it's clearly integrable. But how to show ...
0
votes
1answer
21 views
Sufficient conditions for $z \to \int_\mathbb{R} h(z,x) \,d\mu(x)$ to be analytic
Let $\mu$ be a measure on $\mathbb{R}$ and $G$ an open subset of $\mathbb{C}$. Every function $h \,:\, G\times\mathbb{R} \to \mathbb{C}$ then gives rise to a function $$
F_h \,:\, G \to \mathbb{C} ...
0
votes
0answers
69 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
2
votes
1answer
41 views
Limit of Lebesgue integral in $L_1([-1,1],m)$
Let $([-1,1],\mathcal{M},m)$ be a measurable space in $[-1,1]$ where $m$ is the Lebesgue measure in $\mathbb{R}$ restricted to $[-1,1]$, and $\mathcal{M}$ is the set of $m^*$-measurable subsets of ...
1
vote
1answer
31 views
Limit of Lebesgue integrals in $L_1(\mathbb{R},m)$
Let $g\in L_1(\mathbb{R},m)$ bounded function where $m$ is the Lebesgue measure in $\mathbb{R}$. If
$$\lim_{x\to\pm\infty} g(x)=0,$$
show that for all functions $f\in L_1(\mathbb{R},m)$ we have:
...
5
votes
1answer
88 views
Homogenous measure on the positive real halfline
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
1
vote
1answer
31 views
Lebesgue integral is linear in simple functions.
Let $(X,\mathcal{M},\mu)$ be a measure space. Let $s,t: X\to[0,\infty)$ two simple functions. If $E\in\mathcal{M}$, show that,
$$\int_E (s+t)\,d\mu=\int_E s\,d\mu+\int_E t\,d\mu$$
Attempt:
...
2
votes
1answer
28 views
measure of the image of the the unit open disc by a holomorphic map
I found the following interesting exercice in a textbook:
Let $f$: $\Bbb E \to \mathbb{C}$ be a holomorphic and injective map ('Schlicht function'), where $\Bbb E=\{z \in \Bbb C:|z|<1\}$. ...
-2
votes
1answer
80 views
Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$
Let $(X,\mathcal{M},\mu)$ be a measure space. We say that $\{f_n\}$ converges to $f$ in measure if, for any $\epsilon>0$
$$ \lim\limits_{n \to \infty} \mu\Big(\{x\in X: |f_n(x)-f(x)| \ge ...
-4
votes
0answers
34 views
Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I. Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I need to solve the below.
(1)Show $\mu(\{x\in X| ...
3
votes
1answer
44 views
Measurability of a function defined on a product measure space, and related to a measurable function
Let $ (X,\mu) $ be a standard measure space - so that we may assume that $X$ is the unit interval $[0,1]$ with the Borel $\sigma$-algebra. Consider $X \times X$ with the product measure $\mu \times ...
2
votes
1answer
47 views
Measurable functions on product measures
Let $ (X,\mu) $ be a measure space, and consider $X \times X$ with the product measure $\mu \times \mu $. Consider two functions $f$ and $g$ defined on $X \times X$ such that:
$f$ is measurable.
For ...
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 ...
2
votes
1answer
32 views
Lebesgue Integral defined on infinite measure
Royden's Real Analysis Question: Let {$a_n$} be a sequence of nonnegative real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. I want to show that ...
2
votes
1answer
52 views
Verifying Fatou's Lemma
Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all ...
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.
1
vote
0answers
75 views
Are continuous functions strongly measurable?
Measure theory is still quite new to me, and I'm a bit confused about the following.
Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
2
votes
1answer
34 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
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 ...
1
vote
1answer
48 views
$f_n\to f $ in $L^1$ $\implies$ $\sqrt{f_n}\to\sqrt{f}$ in $L^2$?
Suppose that $\{f_n\}$ is a sequence of measurable functions converging to $f$ in $L^1(\mathbb{R}^n)$. Is it true that $\sqrt{f_n}$ converges to $\sqrt{f}$ in $L^2(\mathbb{R}^n)$?
If this is true ...
1
vote
1answer
148 views
Lebesgue Integral but not a Riemann integral
Is it possible for a function to be a Lebesgue integral, but not a Riemann integral?
After the comments below I realize my question was not a good one. Thank you.
This is my edited version:
Let $f$ ...
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 ...
2
votes
1answer
47 views
Convergence and Lebesgue Integration
I came across this question in a textbook on introductory Lebesgue Integration. I have been teaching myself this material but was unsure of how to do the following question:
Let $(g_n)$ be a sequence ...
1
vote
1answer
32 views
Measurable if and only if absolutely convergent
Let $N$ be the set of natural numbers, $M = 2^N$, and $c$ the counting measure defined by setting $c(E)$ equal to the number of points in $E$ if $E$ is finite and $\infty$ if $E$ is an infinite set. ...
2
votes
1answer
91 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
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
votes
1answer
34 views
Average of a Lebesgue Measurable Function
Say I define an indicator function over a set E and demand that this function have an average of 1/2. It is apparently sufficient to ask that the measure m(E) = 1/2. Why is this? I understood ...
-2
votes
1answer
37 views
Question on Lebesgue Integration
Is it true that a function that is constant except at countably many points is Lebesgue integrable?
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)$ ...
0
votes
2answers
40 views
Real and Imaginary parts of Lebesgue integral in L1 (folland's real analysis)
I am working through Folland's Real Analysis and have a question about a proposition on functions in L1. Prop (2.22) states:
given $f \in L^1$, then $|\int f| \leq \int|f|$
In the proof, for when f ...
2
votes
2answers
70 views
Show $\int_X f d\nu = \int_X fgd\mu$ if $\nu(E)=\int_E g d\mu$ .
$f$ and $g$ are both non-negative functions where the integral of non-negative function is defined as the supremum over all simple functions dominated by the non-negative function.
Would going ...
0
votes
1answer
49 views
Every Borel measure can be written as a Lebesgue-Stieltjes measure?
I was thinking about this and I could not get to an answer. To illustrate my point, assume I have a random variable $X:\Omega\rightarrow\mathbb{R}$ and a measure $\mu$ and I want to compute its ...
-2
votes
2answers
63 views
Combinations of integrable functions
If $f$ and $g$ are integrable functions and real-valued on $(X,M,\mu)$ , which assertion is correct?
$fg\in L^1 (\mu)$
$fg\in L^2 (\mu)$
$\sqrt{f^2 +g^2}\in L^1 (\mu)$
None of ...
2
votes
0answers
63 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 ...
0
votes
0answers
28 views
Question about probability measures on the real line [closed]
http://www2.imperial.ac.uk/~boz/M34P6/P11_2.pdf
Dear comrades. I am struggling with Ex 1.4(i) on here. I think that I have proved that the measures $\mu$, $\nu$, $\lambda$ are all equivalent, in the ...
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
0answers
39 views
Is the Lebesgue integral essentially an inner measure of some kind?
Consider a non-negative Lebesgue-integrable function $f : X \rightarrow \mathbb{R}$, where $X$ is a measure space, and let $F = \{(x,y)|x \in X, y \in [0,f(x)]\}$. Can the Lebesgue integral of $f$ be ...
1
vote
3answers
55 views
How should I calculate the Lebesgue integral of logarithm function from zero to infinity?
Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable?
If yes, how can I calculate it?
1
vote
1answer
62 views
Question about integration (related to uniform integrability)
Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that
$ \int |f| \chi(|f| ...
7
votes
2answers
148 views
Measure theory questions
i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim ...
