Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
40,306
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Show that if $\mu(E)=0$ then $\int_E fd\mu=0$
Suppose $(X,S,μ)$ is a measure space, $f$ is a non-negative measurable function, $E∈S$
, and $μ(E)=0$. I want to show that $\int_E fd\mu=0$
I know that $\int_{X} fd\mu = \sup\left\{\int s d\mu : s \...
- 9
2
votes
0
answers
20
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Sum of $X_k$ with $\mathbb{P}(X_k = 1) = 1/2 + 1/k$ and $\mathbb{P}(X_k = -1) = 1/2 - 1/k$ independently
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = 1/2 + 1/k,
\\
\mathbb{P}(X_k = -1) & = 1/2 - 1/k
\end{align}
for each $k \ge 1$.
...
- 722
2
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31
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Is it possible that $E(E(Y|\mathcal{F}_S)|\mathcal{F}_T) $ is not $\mathcal{F}_{S \wedge T}$-measurable?
Assume you have a filtered probability space $(\Omega, \mathcal{A},P)$, two stopping-times S, T, and an integrable random variable $Y$.
In this post and answer:
Conditional Expectation based on ...
- 1,446
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0
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12
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Confusion about regular conditional distribution of $Y$ given $X$ in Klenke
In Klenke's Probability Theory A Comprehensive Course, the following definition of regular conditional distribution of $Y$ given $X$ is given. I am confused in particular by the last equality:
$$
\...
- 3,082
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votes
0
answers
18
views
What is the relationship between a degenerate limiting distribution and its scaled non-degenerate distribution?
In my statistics class, we are covering modes of convergence: a.s., $L_p$, probability, weak convergence of measure. To motivate the limiting theorems that we will be covering next, my professor ...
- 117
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34
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Lebesgue's Differentiation Theorem and "regularity" of functions in $L(X)$
Let $L(X)$ be the space of Lebesgue integrable functions in $X\subseteq\mathbb{R}^n,$ wrt the Lebesgue Measure. Lebesgue's Differentiation Theorem establishes that the average value of $f\in L(X)$ in ...
- 186
1
vote
1
answer
69
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How is the integral of a simple function well-defined in Folland?
I am reading Folland's real analysis text, section 2.2 on integration of nonnegative functions. I am stuck at the definition of the integral of a simple function and how to show it is well-defined. ...
- 1,035
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votes
2
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29
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Proof that a random variable is constant over certain sets in the sample space
Let $(\Omega, \mathcal{F}, \Pr)$ be a probability space and consider a random element $X: (\Omega, \mathcal{E}) \to (\mathcal{X}, \mathcal{E})$. The sigma-algebra generated by $X$ is
$$\sigma(X) = \{ ...
- 3,082
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72
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Use of the axiom of dependent choice in a standard proof of the Radon–Nikodym theorem
In both this mathoverflow post and this paper, the set theory and logic specialist Robert Solovay indicated that the proof of Radon–Nikodym theorem he read in Halmos' measure theory book used the ...
1
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0
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28
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Question about the Weak-* Limit of the Incremental Ratio
I am studying Michael E. Taylor's book Measure Theory and Integration and revising Chapter 13 on Radon Measures.
Let $ f \in L^\infty(\mathbb{R}) $. Show that the following are equivalent:
(a) $ \...
1
vote
1
answer
38
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$\int_{[0,t]} |f(s)| \, ds \leq \int_{[0,\infty]}e^{M(t-s)} \operatorname{sup}_{s\geq 0} |f(s)|e^{-Ms} \, ds $? Measure theory inequality
For any Lebesgue measurable function $f$, and $M>0$, if $\operatorname{sup}_{s\geq 0}|f(s)|e^{-Ms}<\infty$. I wonder if the following inequality hold $$\int_{[0,t]} |f(s)| \, ds \leq \int_{[0,\...
- 1,768
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14
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Example of a conditional probability distribution that is not regular
In Theory of Statistics (Appendix B.3.1) by M. Schervish, the notions of conditional probability and conditional distribution given a sigma-algebra is defined as follows:
Definition B.29 Let $(S, \...
- 3,082
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25
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A condition that makes outer-measure finitely additive [closed]
It is known that Outer-measure fails to be finitely additive.I am trying to understand the logic behind some conditions that make it finitely
additive. Let $$A,B \subseteq \mathbb{R}$$. Assume that $$...
- 17
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39
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Does every $\sigma$-algebra have a minimal set? [duplicate]
If $(X,\mathcal S)$ is a measurable space does $\mathcal S$ necessarily have a minimal set? More precisely a non-empty set $A\in\mathcal S$ such that
$$B\subsetneq A~,~B\in\mathcal S \implies B=\...
- 69
3
votes
1
answer
21
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Limit of sequence of discrete measures is discrete
Fix $N\in\mathbb{N}$ and consider a sequence of discrete probability measures $\mu_n := \frac1N\sum_{i=1}^N \delta_{x_{i,n}}$, where the points $x_{1,n},\ldots, x_{N,n} \in\mathbb{R}^{d}$. By weak-* ...
- 613
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vote
1
answer
16
views
Do positive functionals on a self-dual cone extend to the whole space?
Let $\mathcal H$ be a Hilbert space with self-dual cone $\mathcal H^+ = \{\,\xi\in\mathcal H \mid \langle\xi|\mathcal H^+\rangle \ge 0\,\}$. Suppose we have a function $f\colon \mathcal H^+ \to [0,+\...
- 736
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votes
1
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59
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Topological Markov chains: how does this computation of 'entropy' work?
Everything I am saying is from Parry's paper 'Intrinsic Markov chains'. I try to summarise the ideas there for my own clarity before giving the question. We have an integer $s$ and the finite set $\...
- 1,649
1
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0
answers
26
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How to under stand this measure estimate in infinity dimension space $[-1,1]^\mathbb{Z}$
Lemma 4.1. Let $(V_n)$ be random in [-1,1]. Then, except on a set of small measure in [-1,1]$^\mathbb{Z}$,
the following holds
(4.1)
$$\left\|\sum'\ell_nV_n\right\|\geqslant\delta\prod_n(1+\ell^2n^4)^{...
- 111
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17
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Nummelin (1984) Existence of Cycles of Irriducible Kernel
I am doing my second read on Nummelin's 1984 book "General Irriducible Markov Chains and Non-negative Operators" available for instance here https://www.cambridge.org/core/books/general-...
- 2,722
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0
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21
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Almost sure convergence of random variables; dependence on $\omega$
Let us for example take the strong law of large numbers which states for $X_1,...,X_n$ defined on a common probability space the quantity
$$
\lim_{n\to \infty}\left|\frac{1}{n}\sum_{i=1}^n X_i(\omega)-...
- 336
1
vote
1
answer
59
views
$X\cup Y$ dense in $U$ open implies existence of $r\in \partial X\cap \partial Y$
Suppose $U\subset \mathbb{R}$ is an open interval and let $X,Y\subset U$ disjoint subsets that each have positive measure. Furthermore, $X\cup Y$ is dense in $U$. Show that there exists $u\in \partial ...
- 95
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votes
1
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76
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Null sets, zero measure sets, and complete probability spaces
Let $(\Omega,\mathscr{F},\mu)$ denote a probability space, and $\mathscr{F}$ a Borel $\sigma$-field of subsets of $\Omega$. I am interested in the interplay between the following two definitions of ...
- 1,020
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0
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16
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$\cup_n F_n=\mathbb{R}^2$, then $F_n$ is dense in $S$
This is from the 2011 Miklos competition:
Let $F_1,F_2,...$ be Borel measurable sets on the plane whose union is the whole plane. Prove there is a natural number $n$ and a circle $S$ such that $S\cap ...
- 2,412
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13
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Proving that a lambda-system generated from a pi-system is closed under countable union
I am having difficulty finding that lambda(I) is closed under countable unions, where I is a pi-system.
I am trying to prove lambda(I) is closed under countable unions because I need to prove it is a ...
3
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1
answer
45
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How can we know the $f$-invariant measures?
I was reading the answer to the following question here What are the $f$- invariant measures?:
Let $X$ be the unit circle in $\mathbb R^2.$ Let $A$ be a $2\times 2$ matrix with real entries, ...
- 169
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1
answer
51
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If $f \in L^1(\mathbb{R})$ is it true that $\lim_{x\rightarrow \infty} |f(x)| = 0$?
My question is pretty much what it is in the title. My intuition tells me that it should be true because in other case we could always find intervals in which the function is greater than 0 and this ...
- 855
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votes
1
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56
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Question on Tonelli theorem for series.
I am thinking to proove tonelli theorem for series using monotone convergence theorem.
$$\sum_{j\geq1}\sum_{i\geq1} a_{ij}=\sum_{i\geq 1}\sum_{j\geq1} a_{ij}.$$
Where $a_{ij}\geq0$.
I am thinking to ...
- 662
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17
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Measurability of conditional quotient of measurable functions
Let $(X,\mathcal A)$ be a measurable space and let $\mathcal B$ denote the Borel $\sigma$-algebra on the real interval $[0,1]$. For two measurable functions $f,g~\colon X \to [0,1]$, define the ...
- 71
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1
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44
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Exercise on Borel-measurability
Let $A$, $B \in \mathcal{B}(\mathbb{R})$, $y_0 \in B$ be an accumulation point for $B$, and let $f: A \times B \to [-\infty, +\infty]$ be a $\mathcal{B}(A \times B)$-measurable function. Assume that ...
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$\mathcal{A}=\{A \subset \mathbb{R}^{d} : A \ \text{is dense in} \ \mathbb{R}^{d} \ or \ \mathbb{R}^{d}-A \ \text{is dense in} \ \mathbb{R}^{d}\}$
Consider $\mathcal{A}=\{A \subset \mathbb{R}^{d} : A \ \text{is dense in} \ \mathbb{R}^{d} \ or \ \mathbb{R}^{d}-A \ \text{is dense in} \ \mathbb{R}^{d}\}$
I need to check whether $\mathcal{A}$ is a $\...
- 89
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0
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40
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Understanding dense subset of $[0,1]$ with Lebesgue measure $\epsilon>0?$
When asked to find a dense subset of $[0,1]\subset\mathbb{R}$ with Lebesgue measure $\epsilon>0,$ there exist many solutions one may find; however, there is one which I have never understood ...
- 1,019
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votes
1
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59
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Examples of *uncountable* null and meagre sets that are not rare
I'm trying to teach myself real analysis, and I'm trying to figure out the various notions of "small" sets. My current understanding lead me to the following Euler diagram:
For the other ...
- 143
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32
views
Weak convergence and approximations of the Dirac Delta [closed]
Suppose
\begin{align*}
a_n &\to \delta_0 \text{ as a distribution}, \\
f_n &\stackrel{\ast}{\rightharpoonup} f \text{ in $L^\infty$} \\
f_n \ast a_n &\stackrel{\ast}{\rightharpoonup} g \...
- 8,320
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votes
1
answer
53
views
Domination by product measure
I have come across something in Grimmets percolation book, that i don't really know to prove. He states around (7.64), that if $Y_i, i \in S$ are $\{0,1\}$ valued variables for a countable set $S$ ...
- 51
2
votes
1
answer
46
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Representation and the corresponding spectral measure
I'm reading a chapter about the relation between Hilbert space and quantum mechanics and got stuck at unclear correspondence.
Let $\mathcal H$ be a Hilbert space, $V$ be $\mathbb{R^{1,d-1}}$ and $U$ ...
-3
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0
answers
14
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Lebesgue measure of the set of numbers with the digits of their decimal part uniformly distributed. [closed]
For $x\in\mathbb{R}$, let $\{x_n\}$ be the sequence of digits appearing after its decimal point, i.e., $x_n=\lfloor10^nx\rfloor\ \mathrm{mod}\ 10$.
Let $p_{d,n}(x)=\frac{\sum_{i=1}^n[x_i=d]}{n}$, and ...
0
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0
answers
25
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On the uniqueness of Caratheodory Extension Theorem when using semi-rings
Suppose that $X$ is a set, $J$ is a semi-ring with respect to $X$, $f$ is a pre-measure defined with respect to $J$, by Caratheodory's Extension Theorem there exist a measure $\mu$ such that $\mu$ ...
- 85
1
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0
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13
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On the absolute continuity and singularity of Patterson-Sullivan currents
I was reading a paper by Alex Furman on the space of metric structures. Let $\Gamma$ be a non-elementary hyperbolic groups and $D_{\Gamma}$ be the space of metrics on $\Gamma$ that is hyperbolic, $\...
- 733
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votes
1
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66
views
Other invariant measures than Lebesgue measure?
Consider a rational rotation of the circle. What are other invariant measures different than the Lebesgue measure?
Any hints will be greatly appreciated!
- 169
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What is the invariant volume form in a surface of Minkowski space
I'm reading Quantum Fields and Strings: A Course for Mathematicians (001) and
ran into a problem.
In p379,
$V=\mathbb{R}^{1,d-1}$,
$V_{space}=\lbrace v\in V\vert v^2<0\rbrace$,
$\overline V$ is ...
0
votes
1
answer
37
views
Extreme points in the set of central probability measures on a group
Let $G$ be a finite group and $CS(G)$ the set of probability measures that are invariant under the adjoint action, so $\varphi\in CS(G)$ exactly when for all $g,\,t\in G$):
$$\varphi(t^{-1}gt)=\varphi(...
- 7,831
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29
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If $\mu_n {\rightharpoonup}\mu$, then under what conditions $|\mu_n - \mu | {\rightharpoonup} 0$?
Let $(X,\mathcal{B})$ be a measurable space where $X \subseteq \mathbb{R}^d$,compact, $\mathcal{B}$ is the Borel sigma algebra, $\mu$ is a finite signed measure.
Let $\left\langle \mu_n\right\rangle_{...
- 1,491
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vote
1
answer
77
views
Does $\int f_n d\mu_n \rightarrow 0$, if $f_n \rightarrow 0$ p.w. and $\langle \mu_n \rangle$ converges to a finite measure?
Let $(X,\mathcal{B},\mu)$ be a measure space where $X \subseteq \mathbb{R}^d$,compact, $\mathcal{B}$ is the Borel sigma algebra, $\mu$ is a finite measure which is absolutely continuous wrt the ...
- 1,491
1
vote
0
answers
38
views
Is the last step in the video necessary to prove exercise 1.4.1 in Durrett 1.4?
In Durrett's Probability: Theory and Examples exercise 1.4.1,i.e., show that if $f\geq 0$ and $\int f d\mu=0$ then $f=0$ a.e.
It's easy to show that for any $\epsilon>0$, $\mu(\{x:f(x)>\epsilon\}...
- 529
0
votes
0
answers
18
views
Radon-Nikodym derivative of posterior wrt prior
Suppose $X \sim \pi_X$ and the RV $Y$ is obtained from $X$ with some noise. Note that $X$ may have continuous or discrete distribution, and $Y|X$ may have continuous or discrete distribtuion.
Now, I ...
- 707
3
votes
1
answer
110
views
Clarification on a proof of the Skorokhod representation theorem
(Skorokhod's representation theorem): Let ${X_1,X_2,\dots}$ be a sequence of real random variables, and $X$ a further random variable. Then ${X_n}$ converges in distribution to ${X}$ if and only if, ...
- 1,121
0
votes
1
answer
34
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Decaying functions on positive reals are Laplace transforms of signed measures
Given any function $f(t)$ that decays on $[0, \infty)$ to zero (but is not necessarily monotonic), is it possible to show that it can be represented as the Laplace transform of a signed Borel measure $...
3
votes
1
answer
87
views
$X$ is measurable w.r.t $\sigma(X_1,X_2)$, then $X=f(X_1,X_2)$ for some measurable $f$?
It is relatively easy to show that $X$ must be a function of $X_1,X_2$. One just need to verify that $X$ must be constant on any level set $\{\omega|(X_1(\omega),X_2(\omega))=(a,b)\}$.
It would be ...
- 92
1
vote
0
answers
31
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Lebesgue integral via Product-measure of Lebegue-Measures
I got a question about product measures. I found an interesting statement in the Book "Taylor, Introduction to Measure and Integration" which I want to quote:
"Suppose ($\Omega, \...
- 113
3
votes
0
answers
38
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Strictly increasing family of sets $\mathcal{E_j}$ which consists of countable unions of sets in $\mathcal{E}_{j-1}$ or their complements
In Folland's Real Analysis, Section 1.6, it says
Our characterization of the $\sigma$-algebra
$\mathcal{M}(\mathcal{E})$ generated by a family $\mathcal{E} \subset
\mathcal{P}(X)$ is nonconstructive,...
- 476