Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
23,131 questions
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upper bound of measures in TV norm.
I have a tight sequence of finite measures $v_i$ on a complete separable metric space, could anyone tell me how to get any upper bound for the object: $\lim\sup\limits_{i\to\infty}\frac{1}{i}\sum\...
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2answers
34 views
Set and Set Complement of Uniform Distribution on [0,1]
This questions first came to mind a few years ago when I was taking a course on real analysis as an undergraduate. I posed it to my instructor but he did not know a means of solving my inquiry. But to ...
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A question about L1 function
Suppose $f\in L^1(\mathbb{R}^d)$ (that is, $f$ is real or complex valued and the Lebesgue measure $\int|f|<\infty$), and suppose $\lambda>0$ and $1\leq n\leq d$. Define $g$ as: for any $x\in\...
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1answer
22 views
Show that finitely many $A_n$ will occur
I tried this problem quite a bit but went nowhere. I wish to solve this using set-theoretic algebra.
Problem statement
Let $A_n$, $n \geq 1$ be a sequence of events such that $ \mathbb P(A_n) \to 0$ ...
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1answer
29 views
Is this “inverse” statement of Fubini’s theorem true?
Consider $f:\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}\to\mathbb{R}$ , Fubini’s theorem says if $f$ is Lebesgue integrable than we can integrate first over $x$ and then over $y$ and get the same value as ...
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1answer
10 views
Are $L_p$ norm and discrete $L_p$ norm comparable?
Are there any estimates on how a $L_p$ norm (say for a compact set in $\mathbb{R}$) is related to a discrete $L_p$ norm, where we could for example consider the Jackson integral on this compact set. ...
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18 views
Absolutely continuous measure and equality to $0$
In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement
If $\mu$ and $\nu$ are two probability ...
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1answer
14 views
Density of “opposite” measure
Let $\mathbb{P}$ be a complete probability measure on the measurable space $(\Omega,\mathcal{F})$. Define the measure $\mathbb{Q}$ on measurable sets $A \in \mathcal{F}$ by
$$
\mathbb{Q}(A)\triangleq ...
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1answer
18 views
Projection of Measurable set of positive measure.
Let $E \subseteq \mathbb{R}^2$ is measurable with positive Lebesgue measure. Let $E_1, E_2 $ be the projection of $E$ on x-axis and y- axis respectively. Can we say that $E_1, E_2 \subseteq \mathbb{R}...
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1answer
12 views
Definition of closed Ideal in the space of finite measures.
Let $M(\mathbb{R})$ be the space of finite signed measure on $(\mathbb{R})$. What is the norm defined on this space? From theorem 1.3.5 in Rudin Fourier analysis on groups. Interscience Tracts in Pure ...
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Weak star convergence of Borel probability measures on a metric space
Let $(X,\rho)$ be a compact metric space and let $P(X)$ be the set of Borel probability measures on the Borel $\sigma$-algebra of $X$. Suppose $\mu_n,\mu \in P(X)$ for $n \in \mathbb{N}$ such that $\...
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The function $φ = \sum _{k=1}^∞ log k \mathcal{X}_{I_k}$ is integrable with relation to Gauss measure?
Prove that for Lebesgue-almost every $x ∈[0,1]$, the geometric mean of the integer
numbers $a_1,...,a_n,...$ in the continued fraction expansion of $x$ converges to some
real number: in other words, ...
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1answer
25 views
Fundamental theorem of calculus for function composition of Lipschitz functions
Let $\Omega$ be a bounded open subset with $C^2$ boundary, $f:\mathbb{R}^d\to \mathbb{R}$ be a Lipschitz function, and $u,v:\Omega\to \mathbb{R}^d$ be measurable functions such that $u_i,v_i\in L^2(\...
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32 views
Is it meaningful to order infinite subsets of $\mathbb{N}$ by probabilistic density?
This is based on a recent question where is was pointed out that the probability of finding a prime tends to 0 as you move into larger and larger numbers.
Conversely the probability of finding a ...
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0answers
11 views
If $I\subseteq\mathbb R$, how exactly is the Lebesgue-Stieltjes measure on $\mathcal B(I)$ associated to a function on $I$ defined?
Let $E$ be a $\mathbb R$-Banach space and $g:[0,\infty)\to E$ be right-continuous and of bounded variation.
How is the Lebesgue-Stieltjes measure $\mu$ associated with $g$ defined?
I'm used to ...
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1answer
21 views
(Use the Vitali covering lemma) to show that the union of closed intervals of real numbers is measurable
I have seen similar questions before but the replies given were less than satisfactory.
Let $\mathcal{F} = \{F_\alpha\}_{\alpha \in J}$ be an arbitrary family of closed intervals of real numbers. Let ...
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0answers
20 views
How to understand the neighborhood of a probability measure?
Let $X_{i}$ be a compact Hausdorff space, $M_{i}$ is the set of probability measures on the Borel subsets of $X_{i}$, $X=\times_{i=1}^{N} X_{i}$, $M=\times_{i=1}^{N} M_{i}$, define $f_{i}(\mu)=\int_{X}...
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1answer
42 views
When is $YE(Y\mid\mathcal{G}) = E(Y^2\mid\mathcal{G})$?
I know this is true when $Y \in \mathcal{G}$, but is there a better (less restrictive) condition that can allow this?
In particular, I want to know when I can say that $E(YE(Y\mid\mathcal{G})) = E(E(...
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2answers
69 views
Proof of $f\in L^1(\mathbb R)$ then $m\{x\mid |f|=\infty \}=0$. Is my proof correct.
Let $f\in L^1(\mathbb R)$. Then $f$ is finite a.e. I did the following proof and my teacher gave me a mark of $0$. What I did is : Let $E=\{x\mid |f(x)|=\infty \}$. We have that$$ \int_E|f|\leq \int_{\...
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1answer
18 views
Alternative definition of sigma algebra generated by random variable
Let $(\Omega, \mathcal F)$ be a measurable space (especially a probability space), and let $Y : \Omega \to \mathbb R$ be measurable. Then we define the sigma algebra generated by $Y$ to be
$$
\sigma(Y)...
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0answers
19 views
How to build measure theoretic conditional expectation for this random walk?
I'm trying to build an understanding of conditional expectations and measure theoretic probability using a sum of Bernoulli coin flips.
Supposedly, I have the random variable $ Y = X_1 + X_2 $, where ...
0
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1answer
34 views
When is a random variable $\mathit{X}:\Omega\rightarrow\mathbb{R}^n$ absolutely continuous?
Given a probability space $\Gamma:=\langle\Omega,\Sigma,P\rangle$ and a random variable $\mathit{X}:\Omega\rightarrow\mathbb{R}^n$, I do not fully understand the definition of absolute continuity of $\...
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0answers
28 views
Change of measure and Girsanov theorem
As far as I can tell, change of measure is just a fancy name for change of variable, i.e. if we have a random variables $X$ then we "change measure" when we define a new variable $Y=f(X)$. Since there ...
2
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1answer
50 views
The set of points where a function $f:[a,b]\to\mathbb R$ is discontinuous is Lebesgue measurable
Prove that the set of points where a function $f:[a,b]\to\mathbb R$ is discontinuous is Lebesgue measurable.
Lebesgue measure of set $A$ means that for any set $S\in\mathbb R,$ $m^*(S)=m^*(A\cap S)+...
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1answer
27 views
Differentials of Measures in the context of Radon Nikodym Derivatives.
I first note that a similar question was asked here: Calculating Radon Nikodym derivative, though the explicit steps used to calculate the derivative were not made clear.
Over measurable space $([0,1]...
6
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2answers
279 views
Non-Borel set in arbitrary metric space
Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous ...
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1answer
13 views
Let $(X , \cal{A}, m)$ be a measure space. Let $f:X \to [0,1]$ be measurable. If $m(X) < \infty$, find $\lim_{n\to\infty} \int f^n \, d m$.
Let $(X , \cal{A}, m)$ be a measure space. Let $f:X \to [0,1]$ be a measurable function. If $m(X) < \infty$, determine $\lim_{n\to\infty} \int f^n \, d m$.
So far I have:
If $f(x) < 1$, then $\...
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1answer
46 views
How to prove that $m^*(A\cap B)=0$? [on hold]
Let $A,B\subset\mathbb R$. Then $m^*(A\cap B)+m^*(A\cup B)\le m^*(A)+m^*(B)$
It holds that $m^*(A\cup B)\le m^*(A)+m^*(B)$ for any $A,B$
hence $A\cap B$ must be the empty set.
how to prove that $(A\...
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1answer
35 views
Support of a Lebesgue-Stieltjes measure [on hold]
Let $G:[0,1]\to[0,1]$ be a distribution function (that is, non-decreasing and right-continuous) with support $\text{supp}(G)$ define as:
$$\text{supp}(G)=\{r\in [0,1]: G(r+\varepsilon)-G(r-\...
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1answer
21 views
Example of countably additive function from a Boolean algebra to [0,infinity] that is not finitely additive
This example is implied to exist by Tao’s undergrad measure theory text, exercise 1.7.4, with the claim that further assuming that the empty set getting mapped to 0 prevents such a function from ...
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1answer
26 views
Every measure on a discrete space is $\Sigma$-finite?
I was trying to solve the exercise of the book "E. Cinlar - Probability and Stochastics" on page 18, No. 3.13 (c) but on the second part I lack ideas.
EX. Let $E$ be a countable set and $\mathcal{E}=...
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1answer
10 views
Example of a saturated measure
We know that every sigma finiye measure is saturated but converse is not true..
To prove the converse part , I need a counter example . Can anyone suggest an example of a saturated measure which is ...
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0answers
34 views
Existence of functionals on $L^0$
Studying a paper about risk measures by F. Delbaen, I bumped into this statement:
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
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1answer
34 views
Show $\left| \{H,T\ \}^{\oplus \mathbb{N} }\right| = \left| \mathbb{R}^{\mathbb{N} }\right|$
Let $\mathbb{R}^{\mathbb{N}} = \{ f: \mathbb{N} \to \mathbb{R}\}$ with the product $\sigma$-algebra $\mathcal{B}^{\otimes \mathbb{N}} = \bigotimes\limits_{n \in \mathbb{N}} \mathcal{B}$, where $\...
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1answer
21 views
Show that the set $A = \bigcup\limits_{k \in \mathbb{Z}}[3^k - 2^{-|k|}, 3^k)$ is Lebesgue measurable and find its measure.
Show that the set $A = \bigcup\limits_{k \in \mathbb{Z}}[3^k - 2^{-|k|}, 3^k)$ is Lebesgue measurable and find its measure.
I'm having trouble with both parts of the question.
First Part:
By ...
2
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1answer
38 views
Lipschitz equivalent definition.
Is Lipschitz continuous equivalent to this definition, f is Lipschitz continuous on [a,b] if for each $\epsilon > 0$ there exists $\delta > 0$ such that for every finite collection of intervals $...
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1answer
44 views
Measures on infinite product of $[0,1]$?
What are the known measures defined on some sigma algebra on infinite product space of $[0,1]$ ?
Is there any measure which is compatible with usual metric space structure of infinite product of $[0,...
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1answer
40 views
Weak Convergence of Probability Measures (Proving integrals converge if measures do)
Note: I am doing this question just for fun, not for hw.
Question: Fix any dense subset $G$ of the unit ball of $C^{0}(M)$. Here $C^0(M)$ refers to the space of continuous functions defined on $M$, a ...
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0answers
16 views
Integration w.r.t. pushforward measure
Problem: Let $\phi \colon [0,1] \to [0,1]$ be a continuous function and let $\mu$ be a Borel probability measure on $[0,1]$. Suppose $\mu(\phi^{-1}(E)) = 0$ for every Borel set $E \subseteq [0,1]$ ...
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1answer
19 views
Conditional Distributions and Rationals
If $U$ is uniformly distributed on $[0, 1]$, what is the conditional distribution of $U$ given that $U$ is rational? Intuitively, it would be uniform, but we cannot have a uniform distribution over a ...
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1answer
50 views
How to see the convergence/divergence of this integral [duplicate]
I have a question, I want to know for which values of $p\in \mathbb{R}$ this integral have finite value:
$$\int_{0}^{\infty} \frac{\sin{x}}{x^p}dx $$
I have shown that for $p=1$ the integral is finite ...
0
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2answers
24 views
Interperting $\bigcap_{J\subset I\\J<\infty}\sigma(\bigcup_{j\in I\setminus J}A_j)$ tail sigma-algebra
Tail $\sigma$-algebra. Let $I$ be a countably infinite index set and let $(\mathscr{A}_i))_{i\in\mathbb{N}}$ be a family of $\sigma$-algebras. Then:
$T((\mathscr{A}_i)_{i\in\mathbb{N}})=\bigcap_{...
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0answers
11 views
measurability of a set-valued map from a game
Let $(\Omega,\Sigma)$ be a measurable space, $A$ be a finite set, $B$ be a compact subset of a separable metric space. Let $\Gamma_1:\Omega\rightrightarrows A$ be a measurable set-valued map, and $\...
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1answer
27 views
how to show that $A_{k-1} \subset 3A_{k} $ where $ A_k $ are iterations of the Cantor set ???
For example
$A_1 = [0,1/3] \cup [2/3 , 1] $
$A_2 = [0, 1/9] \cup[2/9 ,1/3] \cup [2/3 ,7/9] \cup [ 8/9 ,1]$
Note that
$3 (A_2) = A_1 \cup D $ , for some set $D$
In the general case, I want to ...
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1answer
34 views
4
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1answer
44 views
An application of Kolmogorov $0$-$1$ law
I need to approve or to disapprove the following statement :
if $(X_n)_{n \in \mathbb{N}}$ is a sequence of independent random variables and identically distributed, and $(u_n)_{n \in \mathbb{N}}$ is ...
0
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1answer
71 views
For every $\epsilon\gt 0$, $|a-b|<\epsilon $ ,then b=a .
I have done a proof by myself but not sure about it
proof: $|b-a|<\epsilon $
=$a-\epsilon $
0
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0answers
29 views
Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries
I'm studying Lebesgue Measure. I have a problem on proving that Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries
Here is my work so far.
Let $T$ $\mathbb{R}^k \to \mathbb{R}^k$ is an ...
0
votes
1answer
19 views
A necessary condition for measurability?
I have to prove that
If $E \subset \mathbb R$ is measurable and $\mu ^*(E)< \infty$, then for each $\epsilon>0$, $\exists A \subseteq E$ such that $A$ is compact and $\mu^*(E \setminus A)<...
1
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0answers
31 views
The integral of a function is 0 in intervals, so the function is 0 everywhere
I am having trouble solving this problem of my course of measure and integration.
Let $f$ an integrable function on the measure space $\mathbb{R}$, $L$, $\lambda$, where $L,\lambda$ is the Lebesgue-...
