Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
28,057
questions
0
votes
1answer
9 views
Cantor Ternary Set Problem, Ternary Expansions
I'm reading Lebesgue Integration on Euclidean Space by Frank Jones and I'm stuck on this problem, of page 41, problem 15.
Let $C$ the Cantor Ternary Set.
Let $G_{1}=(\frac{1}{3},\frac{2}{3})$, $G_{2}...
0
votes
1answer
16 views
Discrete measure and Lebesgue measurability
According to wikipedia
https://en.wikipedia.org/wiki/Discrete_measure
a driscrite measure is defined in the following way:
Let's consider a real line $\mathbb{R}$. For some (possibly finite) ...
0
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0answers
9 views
$\lambda_f(a)$ (Question 6.40 from Folland's Real Analysis Book)
In his book he defined:
$$ \lambda_f(\alpha)=\mu\left(\left\{x\middle||f(x)|>\alpha\right\}\right)$$
And with that in mind, he later defined in the excercise for some $\mu$-measureable $f$ on $X$:
$...
1
vote
0answers
18 views
Radon-Nikodym derivative for discrete measures
Could you, please, check the following:
Let's consider a real line $\mathbb{R}$ and let's define two measures on it. For some (possibly finite) sequences $s_{1}, s_{2}, \dots$ and $a_{1}, a_{2}, \...
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0answers
22 views
Measure of union of diagonals
Let $\mu$ be a measure on $[0,1]$ which induces the product measure on $[0,1]^n$. Let $\Delta_{i_1,\cdots,i_k}$ be the set of points in $[0,1]^n$ whose co-ordinates of $i_1,i_2,\cdots$ and $i_k^{th}$ ...
1
vote
1answer
35 views
Pass to the limit under the sign of the integral of $f_{n}(x)$ [closed]
I'm trying to investigate following limit:
$$
\lim_{n \to \infty}\int_{0}^{\frac{\pi}{2}}\frac{cos^{n}x}{1 + x^{3}}dx
$$
I have a couple of questions
1. Is it possible to find the limit by the ...
0
votes
2answers
30 views
The set of real numbers whose product is rational is Borel in $\mathbb{R}^2$
Let $A=\{(x,y)|xy\in \mathbb{Q}\}$ be a set in $\mathbb{R}^2$. Show that A is a Borel set, and find its Lebesgue measure $m_2(A)$.
This is an exercise of the chapter of Fubini and Tonelli's theorem,...
1
vote
1answer
49 views
Why isn't it obvious, from the definition of the essential supremum, that $\lvert f \rvert \leq \|f\|_{\infty}, \ \mu-$a.e?
Let $(X,\mathcal{A},\mu)$ be a measure space, and $f \colon X \to \mathbb{C}$ an $\mathcal{A}$-measurable function.
My texbook defines, for essentially bounded functions with respect to $\mu$, i.e., $...
0
votes
1answer
22 views
Density and Outer Measure
If $A \subset \mathbb{R}^{n}$ is Lebesgue measurable, then we see that $x \in A$ is a point of density if
$$
\lim_{r \to 0} \frac{\mu(A \cap B(x,r))}{\mu(B(x,r))} =1
$$
The Lebesgue Density Theorem ...
0
votes
1answer
12 views
Convergence in measure and related inequality
Let $f_n$ be a sequence of bounded measurable functions on some probability space. Let $f_n$ does not converge to $0$ a.e. Does it mean that there exists a set of positive measure $E$ such that $\lim\...
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0answers
20 views
Smallest ring, algebra, $\sigma-$ring and $\sigma-$algebra in P(N) that contains Z:{{n}:n is even}
I need to find the smallest ring, algebra, $\sigma-$ring, and $\sigma-$algebra in P(N) that contain Z:{{n}:n is even}, where P(N) is the power set.
I think that the smallest ring is {$\phi$,{0},{2},{...
1
vote
0answers
19 views
Proving implications between statements relating to Lebesgue-Stieltjes measure
$\mathcal{M}_\mu$ is the domain of the Lebesgue-Stieltjes measure $\mu = \mu_F$, where $F$ is an increasing, right continuous function on the Reals. I'm trying to prove the following:
If $E \subset \...
3
votes
1answer
68 views
$(\int f_1d\mu)^2+\cdots+(\int f_nd\mu)^2\leq(\int \sqrt{f_1^2+\cdots+f_n^2}d\mu)^2$
Let $(X, \mathfrak{B}, \mu)$ be a measurable space, possibly not $\sigma$-finite, and $f_1, \cdots, f_n \colon X\to (-\infty, +\infty)$ be integrable functions on $X$. Does $$(\int f_1d\mu)^2+\cdots+(\...
0
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0answers
23 views
Equivalent notions of uniform integrability
I have seen many questions related to uniform integrability in this forum I though that it might be of interest to some to have yet another set of equivalent notions of uniform integrability. Here is ...
2
votes
1answer
40 views
PDF with a finite second moment converges to zero?
Suppose for a probability density function $p(x)$, $x\in \mathbb{R}$, we have $\int_{-\infty}^{+\infty} x^2p(x)dx < \infty$. Can we claim $\lim_{x\rightarrow\infty} p(x) = 0$?
I think it boils ...
2
votes
0answers
34 views
Are jointly measurable adapted processes relative to natural filtration from right-continous processes, progressively measurable?
Recently i'm studying the book of Stroock and Varadhan - "Multidimensional Diffusion Processes" and i'm trying to solve this exercise :
For the first part i argue in the following way :
"Because ...
3
votes
1answer
76 views
Comparing the sizes of null sets
This question is about comparing the relative sizes of null sets by switching from open covers to open covering sequences (a la strong measure zero sets or microscopic sets). The main question is ...
0
votes
1answer
16 views
Show that pushforward measure is inner regular
Let $X, Y$ be compact Hausdorff spaces, $\tau:Y \to X$ continuous and $\mu^\tau:=\mu \circ \tau^{-1}$ the pushforward measure of the measure $\mu$ which is a inner regular measure. Show that $\mu^\tau$...
0
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0answers
30 views
If $Q$ is a rectangle and if $D\subseteq Q$ is the set of points where $f:Q\rightarrow\Bbb{R}$ is discontinuous then $\int_Q f$ exist if $m(D)=0$
What shown below is a reference from "Analysis on manifolds" by James R. Munkres
So I want discuss
why if $f$ is continuous at $a$ then there exist a rectangle $Q_a$ such that $a\in\overset{Ā°}Q_a$ ...
0
votes
0answers
10 views
Expected value of random variable over a shrinking set. (Left-derivative of superexpectation)
I am working on a proof related to the left-derivative of the superexpectation operator $E_X(x) = E[\max\{X, x\}]$.
Let $X \in L^2(\Omega, \mathcal{F}, P)$ be a random variable. Let $x', x \in \...
4
votes
1answer
44 views
Is $(X, Y)$ always absolutely continuous with respect to $P_X \otimes P_Y$?
Definitions:
Let $X: (\Omega, \mathcal A) \to (\mathbb R, \mathcal B)$ be a random variable on the probability space $(\Omega, \mathcal A, P)$ and define its distribution as the probability measure $...
0
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0answers
42 views
If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ ; is $f$ zero a.e?
If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ for $n \in \mathbb{N}$ ; is $f$ zero a.e?
Here $L^1[0,1]$ means the set of measurable functions $g : \mathbb{R} \rightarrow \mathbb{...
1
vote
0answers
12 views
Can we get simultaneous convergence of the integral using simple functions?
Let $Y$ be a nonnegative random random variable on some probability space $X$, and Let $F:[0,\infty) \to \mathbb R$ be a continuous function.
Do there always exist simple functions $Y_n \ge 0$ on $...
0
votes
0answers
15 views
$ \int_{T}^{*}{\psi (t) d\mu(t)}=\int_{T}{\phi (t) d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...
0
votes
1answer
30 views
Sigma algebra generated by product of random variable
This question relates to this.
Let $X,Y$ be two (real) random variables on $(\Omega,F)$. It was stated that
$$\sigma(XY)\subseteq \sigma(X,Y).$$
The argument used is as follows: let $f(x,y)=xy$ be a ...
2
votes
0answers
9 views
$ \int_{E}^{*}{h(t,f_n(t)d\mu(t)}=\int_{E}{a_0\big(t,f_n(t)\big)d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space, and $E$ a convex subset of a Hausdorff topological vector space. Any function from $T\times E$ into
$( - \infty, + \infty]$ is be called ...
0
votes
1answer
18 views
How to prove this function is L-continuous almost-everywhere?
Definition: Let $(X, \mathcal{M}, \mu)$ a measure space. Some property P (in this case, the continuity) is said to be satisfied almost everywhere in X if there ...
0
votes
1answer
21 views
Inclusion of sigma-algebra generated by a sequence of random variables
Let $x_n$ be a sequence of random variables on $(\Omega,F,P)$ and $a_n$ be a sequence of real numbers. For any $1\leq i<k<\infty$, I want to evaluate if
$$\sigma(a_mx_m: i\leq m\leq k)\subseteq \...
0
votes
0answers
29 views
The boundary of a disc has zero measure
Consider that a subset $A$ of $\mathbb{R}^{n}$ has measure $0$ if for every $\epsilon > 0$, there is a cover $\{U_{1}, U_{2}, \cdots \}$ of $A$ by closed (or open) rectangles such that
$$\sum_{i = ...
3
votes
1answer
32 views
there exists a sequence of simple functions $\{g_n\}$ bounded in $L^1$, such that $f_n-g_n\to 0 \text{ a.e and in } L^1.$
Let $(E,\mathcal{A},\mu)$ be a finite measure space, and $\{f_n\}$ be a sequence bounded in $L^1$.
Why does there exist a sequence of simple functions $\{g_n\}$ bounded in $L^1$, such that:
$$
f_n-...
0
votes
1answer
44 views
Can we say that: $\|g\|_\infty<M $?
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset \mathcal{L}^2$, such that:
$$
\forall n\geq 1~:~|f_n|\leq M~~a.e\qquad (1)
$$
and
$$
\exists g\in\mathcal{L}^1, \text{such that ...
4
votes
0answers
23 views
I can say that : according to the previous lemma we have: $ \| u_k \|_2\leq 2 \| F_k (f_n ^ k) \|_2, \qquad \forall n \geq 1.$
Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $ X $ be a Banach space and $ H $ a Hilbert space.
For $ t \in E $, we set $ F_a (f)(t) = f (t) 1_{\| f \| \leq a} (t)$
Lemma: Let $ \{...
0
votes
0answers
35 views
All bounded sequence of $\mathcal{L}^2$ is weakly relatively compact of?
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset \mathcal{L}^2$, such that:
$$
\sup_n \int_{E}{|f_n(t)|^2d\mu(t)}<+\infty
$$
Why $\{f_n\}$ is weakly relatively compact of $\...
0
votes
0answers
29 views
$\sigma$-algebra generated by a finite family of sets
The question is the exercise 1.7 of "Measure theory and probability theory" by Krishna and Soumendra.
Let $\mathcal{B}=\{B_1, ..., B_k\}$ with $B_i \subseteq \Omega$ and $\bigcup_{i=1}^kB_i=\Omega$. ...
0
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0answers
25 views
Reference to Lebesgue space on manifolds [closed]
I am looking for a reference to the Lebesgue space $L^p(M)$, where $M$ is a manifold.
2
votes
0answers
28 views
Finding adjoint operator
Let $X, Y$ be compact Hausdorff spaces. Let $\tau:Y\to X$ be continuous and $A \in \mathcal B(C(X),C(Y))$ the operator given by $Af:=f\circ \tau$.
Show that the adjoint operator $A' \in \mathcal B(...
0
votes
0answers
15 views
Criterion for function to be measurable on product space
I was wondering if there is some sort of go to criterion to check if a function is measurable on a product space. Consider for example the following situation. Let $f: \Omega_1 \times \Omega_2 \to \...
1
vote
0answers
31 views
Asymptotically vanishing probabilities
Let $X_1,\ldots,X_n$ be a sequence of independent and identically distributed random variables, with joint probability measure $\mu^{(n)}$. Let $f_n:\mathbb{R}^n \to \mathbb{R}$ and $g_n:\mathbb{R}^n \...
0
votes
1answer
13 views
sigma algebra created by partition
So there is a countable (disjoint) partition of $\Omega = \bigcup_{i \in \mathbb{N}}B_i$ and now I'm interested in the $\sigma$-algebra created by this partition $\sigma(\{B_i:i\in\mathbb{N}\})$.
I've ...
0
votes
0answers
37 views
How to prove this equation with complex number?
I am stuck to proof this ineuqality.
$|\frac{z}{z^2-1}-\frac{z}{z^2-4}|\leq\frac{\pi}{2\eta}$ with $\eta=Im z$ (Imaginary part of complex number). $Im z>0$
Here z is complex and $\epsilon\in(0,1)...
0
votes
1answer
24 views
Why does linear combination of dirichlet function is not a step functionļ¼
I know that Dirichlet function is not Riemann integrable but only Lebesgue integrable which has measure zero. I am having issue to understand the example that a simple function which is not step ...
0
votes
1answer
34 views
Can we say that $ \frac{1}{n}\sum_{i=1}^{n}{f_i(t)\to 0 }~~\text{a.e}$?
Let $(E,\mathcal {A},\mu)$ be a finite measure space and $\{f_n\}$ be a sequence bounded in $L^1$, such that:
$$
f_n(t)\to 0 ~~\text{a.e and in } L^1
$$
Can we say that
$$
\frac{1}{n}\sum_{i=1}^{n}{...
1
vote
0answers
19 views
Sigma-algebra generated by a sequence with zeros
Let $\{x_i\}_{i\in\mathbb{N}}$ be a sequence of zero mean random variables on a probability space $(\Omega, F, P)$. Suppose that there is some $i_0>0$ such that $x_i=0$ for all $i\geq i_0$.
Does ...
1
vote
0answers
27 views
Doubt about the computation of the volume of a sphere
Consider the measure $d\Omega_{n+1}$ as a invariant measure induced by the Lebesgue measure on the unit sphere $\mathbb{S}\subset \mathbb{R}^{n+1}$. I want to calculate $d\Omega_{n+1}(\mathbb{S})$. I ...
2
votes
2answers
73 views
Is BanachāTarski paradox false without axiom of choice? [duplicate]
I know that you need axiom of choice to prove BanachāTarski paradox. But what happens with paradox when we remove axiom of choice? Does theorem become false? Or is there just no proof of it without ...
0
votes
0answers
24 views
Reference for the Dunford-Pettis Theorem.
Does anyone have a book reference for the Dunford-Pettis theorem as stated in the Wikipedia article https://en.wikipedia.org/wiki/Uniform_integrability.
4
votes
0answers
53 views
Hilbert space version of the notion of conditionally weakly-mixing functions.
$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\mr}{\mathscr}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\C}{\mathbf C}$
...
3
votes
0answers
49 views
Lp Space: an intuitive understanding.
Would someone be willing to give me an intuitive understanding of the Lp space? I have several analysis books which (seemingly) approach the topic differently, which confuses me more, and a simple ...
0
votes
0answers
14 views
Question regarding definition of Lebesgue integral
I am reading Measure Theory by Halmos, and I was wondering if someone could help me on the following definition:
If there exists a mean fundamental sequence $\left\{f_{n}\right\}$ of integrable ...
1
vote
1answer
35 views
Is there a missing condition in this statement regarding Borel measurability
I'm self-studying measure theory. I stumbled upon this statement:
If $X \subset \mathbb{R}$ and $f: X \to \mathbb{R}$ is a function, then $f$ is a Borel measurable function if and only if $f^{-1}((...
