Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
23,469 questions
0
votes
1answer
18 views
Approximating the integral of an $L^1$ function by the integral over a finite set.
Let $(\Omega,\mathcal A,\mu)$ be a measure space and let $f\in L^1(\mu)$. Show that for any $\varepsilon>0$, there is an $A\in\mathcal A$ with $\mu(A)<\infty$ and $|\int_A f\ \mathsf d\mu - \int ...
2
votes
1answer
19 views
Since there is conditional probability, is there any thing like conditional measure?
If I have two measure space $X,Y$, with measure $\mu_X$ and $\mu_Y$, the usual way to define a measure for the combined measure space $U=X \times Y$, is the product measure, i.e.,
$$\mu_U = \mu_X \...
1
vote
0answers
15 views
Girsanov THM and Radon-Nikodym derivative
I've been having a hard time to applicate Girsanov theorem with Radon-Nikodym derivative in the demonstration of German-El Karoui-Rochet formule.
I know that $\Pi_0:=S_0\mathbb{Q}^S(S_T\geq K)-K\...
0
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0answers
6 views
Integrating distance function over circle
I'm confused about the following integral. We fix $s\in S^1$:
$$\int_{S^1}d_{S^1}(s,y)dH_{S^1}^{1}(y)$$
I tried to compute it brute force by separating the integral in two parts:
$$\int_{S^1}d_{S^1}...
2
votes
1answer
33 views
A measurable function is an infinite sum of simple measurable functions. Why measurable?
Let $\left(E,\mathcal{E}\right)$ a measurable space and let $f:E\to[0,+\infty]$
a measurable function. Then
$$
f=\sum_{i=1}^{+\infty}a_{i}\chi_{A_{i}}
$$
with $a_{i}\geq0$ and $A_{i}$ measurable.
...
3
votes
1answer
29 views
Stronger version of finite additivity of Legesgue measure
Let $m_*(A)$ denote the Lebesgue outer measure, and when $A$ is measurable, let $m(A)=m_*(A)$ be the Lebesgue measure.
Let $U=\{E_1,\cdots,E_N\}$ be a finite collection of pairwise disjoint Lebesgue ...
1
vote
2answers
58 views
understanding $\theta,\Theta, d\theta, \mu(d\theta)$
Let $(\mathcal S, d)$ be a metric space. let $\{f_\theta: \theta \in \Theta\} $ be a family of Lipschitz functions on $\mathcal S$ and $\mu$ be a probability distribution on $\Theta$. Suppose that $\...
1
vote
0answers
18 views
Product sigma algebra of countable cocountable sets
I want to prove the following proposition:
Let $\Omega $ be an uncountable set and let $$\mathcal F = \{A \subset \Omega: A \textrm{ countable or } A^c \textrm{ countable} \}.$$
If $B \in \...
0
votes
0answers
26 views
Evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$. [on hold]
Let $X,Y$ independent random variables, identically distributed and symmetric about $0$.
Select some probability density function f(.), and evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$.
2
votes
1answer
28 views
Is all random variable finite (almost surely)?
I feel like what I've known are conflicting with each other, so I'd like to post it.
When some prove that if $X_n \to X$ in probability and $Y_n \to Y$ in probability, then $X_nY_n \to XY$ in ...
0
votes
0answers
8 views
What is the disintegration of a measure with respect to another measure
Suppose we have a Borel measure $\mu$ on $\mathbb{R}$ and a Borel measure $\pi$ on $\mathbb{R}^2 = \{(x, y): x \in \mathbb{R} = X, y \in \mathbb{R} = Y\}$. Suppose that the marginal of $\pi$ on $X = \...
0
votes
1answer
16 views
Elements in a sigma algebra generated by a partition
Suppose you have a set $X$. Consider the $\sigma$-algebra $\mathcal{A}:= \sigma\{A_1,A_2,\cdots,A_n\}$, where $\{A_1,A_2,\cdots,A_n\}$ form a partition on $X$. Is it then true that every element $A \...
1
vote
0answers
39 views
If $\theta$ are i.i.d rv over some parameter set, then what is $d\theta$
Could anyone make me understand, or tell me how to understand, what is $d\theta$ and $\mu(d\theta)$, here in page $47$, Theorem $1.1$ in the paper Iterated Random Functions?
Thanks a lot for helping....
2
votes
0answers
30 views
Lp space is a Hom functor
Is there way to see $L^1$ a functor from the category with objects as metric measure spaces $(X,d,\mu)$ and morphisms Lipschitz maps to the category of Banach spaces (or something which has a ...
0
votes
0answers
21 views
Fubini's theorem and the undefinedness.
$\newcommand{\I}[1]{\displaystyle{\int_Y #1}\ \mathrm{d}y}$
The Fubini's theorem in standard form (for example in: Wikipedia, § For integrable functions), asserts that the inner integral (drawing ...
-1
votes
0answers
12 views
What assumption makes the followin true? $ \lim_{h \to 0} \frac{1}{h}\int_{B_R(0)} 1 \wedge \left(f( x + h) - f( x) - ⨏_{B_h(x)} f dz \right) dx = 0 $
What assumptions on the function $f: \mathbb R^N \to \mathbb R^N$ make the following true?
$$ \lim_{h \to 0} \ \ \
\frac{1}{h}\int_{B_R(0)} 1 \wedge \left(f( x + h) - f( x) - \int_{B_h(x)}\!\!\!\!\...
3
votes
1answer
31 views
Prove that $g(x):=\int_0^1f(x,y)dy$ is Borel measurable.
Let $f: [0,1]\times [0,1]\to\mathbb R$ satisfy:
(i) for each $x\in [0,1]$, the function $y\mapsto f(x,y)$ is Riemann integrable on $[0,1]$; and
(ii) for each $y\in [0,1]$, the function $x\...
2
votes
0answers
36 views
Difference between $\mathbb E^x[X_t]$ and $\mathbb E[X_t\mid X_0=x]$?
Let denote $\mathbb P^x\{X_t\in A\}=\mathbb P\{X_t\in A\mid X_0=x\}$ and $\mathbb E^x$ the expectation associated to this measure, i.e. $$\mathbb E^x[X_t]=\int X_t\,\mathrm d \mathbb P^x.$$
I was ...
0
votes
0answers
8 views
Non separability implies $\sigma(W^*)$ not Borel
I am looking for a proof or counter example to the following:
Let $W$ be an inseparable banach space and $W^*$ be its topological dual; then $\sigma(W^*)\neq B(W)$ for $B(W)$ is the Borel sigma ...
0
votes
1answer
22 views
Nonseparability of $l_2(…)$?
Let $X=l_2([0,1])$ be a space of sequences such that $x_n\in [0,1]$ for all $n\in \mathbb{N}$ with the standard norm of $l_2$. How to prove that $X$ is nonseparable?
0
votes
3answers
36 views
measure and countable set [on hold]
how can we prove that if $\mu$ is a probability measure on $(\mathbb{R}^d,B(\mathbb{R}^d))$ then the set $E:=\left\{x \in \mathbb{R}^d,\mu(\left\{x \right\}) \neq 0\right\}$ is countable.
Thank you.
2
votes
1answer
42 views
Function is Baire-1 if and only if these sets are $F_\sigma$
I AM LOOKING FOR A HINT, NOT A FULL SOLUTION, TO THE FOLLOWING PROBLEM:
A function $f: [a,b] \to \mathbb{R}$ is called Baire-1 if it is the pointwise limit of a sequence of continuous functions. ...
1
vote
0answers
41 views
Why can we apply Fubini's theorem here?
Let $f : \mathbb{R^n} \rightarrow \mathbb R$ a measurable function define the function $t \mapsto \lambda_f(t) = \mathcal L\{ x : \vert f(x) \vert > t\}$ where $\mathcal L$ is the Lebesgue measure ...
0
votes
0answers
25 views
Explanation for Oxtoby's proof: a nonempty topological space $X$ is Baire iff player (I) has no winning strategy in the Choquet game
A nonempty topological space $X$ is a Baire space iff player I has no winning strategy in the Choquet game $G_X$.
Oxtoby's proof
I have several questions about this proof.
$(\Leftarrow)$
How can ...
0
votes
1answer
24 views
Showing a function is in $ L^\infty$
Let $(X,\mu)$ a finite measure space and $g\in L^2_\mu$. Suppose there is a constant $K\ge 0$ s.t.
for any measurable set $B$ , $|\left<g,\chi_B\right>|\le K \cdot \mu(B)$, how can I show that ...
0
votes
0answers
35 views
$f$ is measurable if and only if $E_i$ is measurable
Let $f:X\to\mathbb R$ be a simple function in the standard representation, $\displaystyle f=\sum_{i=1}^n\alpha_i\chi_{E_i}, $ where $\alpha_i\neq\alpha_j,E_i\cap E_j=\emptyset$ if $i\neq j$ and $\...
1
vote
0answers
42 views
left uniform continuous function that is not right uniform continuous
Consider the following problem 5 below.
I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of ...
0
votes
0answers
32 views
What is the base measure in measure theory?
I see the term "base measure" used frequently about measures. I do not completely get what that exactly means:
Some examples are:
Let $\cal F$ be the space of all probability density functions ...
0
votes
0answers
16 views
Can a stochastic process be neither adapted to filtration nor previsible?
The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'.
If a process is adapted, it essentially means that the evolution of the ...
3
votes
2answers
38 views
Let $\mu(X)=1$, $0 \leq f \leq k$, and $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.
Let $\mu(X)=1$ for $\mu$ a positive measure. Let $0 \leq f \leq k$ for some $k\in\mathbb{R}$ and let $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.
My attempt:
I tried to expand ...
0
votes
0answers
18 views
Show that every measurable function $f$ is the limit of a sequence of continuous functions converging in measure.
This is a challenging question from my Analysis class: Given a measurable space and a finite regular measure defined on the $\sigma$-algebra, show that every measurable function $f$ is the limit of a ...
0
votes
0answers
40 views
about unmeasurable set on R
Suppose $\{E_n\}$ is a sequence of sets in $\mathbb{R}$, and $E_n$ are pairwise disjoint sets satisfying
$$
m^*(\cup_{n=1}^{\infty}E_n)<\sum_{n=1}^{\infty}m^*E_n.
$$
I want to prove that there ...
0
votes
0answers
10 views
Wave cone of the curl operator
How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as
\begin{equation*}
\Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi ...
0
votes
1answer
25 views
Continuity of Lebesgue integral of integrable function
I'm tackling the following question:
My approach to (a) was:
Let $x \in (0, \infty)$ and consider any sequence $x_j \to x$. We are asked to prove that $v(x_j) \to v(x)$ i.e. that
$$\int_\mathbb R \...
0
votes
1answer
27 views
Compactness of the set of finite Borel measures
Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n \in \mathbb N$. Let $\mathcal M(X)$ denote the space of all finite Borel measures on $X$. Is $\mathcal M(X)$ compact under some commonly ...
1
vote
1answer
37 views
Showing Lebesgue Integral inequalities
Let $f,g: [0,1] \longrightarrow (0,\infty)$ be measurable and $\beta >0$. Assume that $$\int_{0}^{1}g(x)dx = 1.$$ Show that $$1\leq \Bigg(\int_{0}^{1}f(x)^{-\beta}g(x)dx\Bigg)\Bigg( \int_{0}^{1}f(x)...
0
votes
0answers
32 views
Showing differentiability [on hold]
Let $\phi: \mathbb{R} \longrightarrow \mathbb{R}$ be a bounded differentiable function such that $\phi^{'}$ is bounded as well. Let $f \in L^1(\mathbb{R})$. Define $F:\mathbb{R} \longrightarrow \...
0
votes
1answer
23 views
For non $\sigma$-finite $d\mu$, integrable $f$, and $A_f = \{ f > 0 \}$, show that $\mathcal{V}(E) = \int_E \chi_{A_f}(x) \, d\mu$ is $\sigma$-finite.
Let $f \ge 0$ be integrable for a measure $d\mu$ e.g. $\int f \, d\mu < \infty$.
Here $d\mu$ will not in general be $\sigma$-finite.
Let $A_f = \{ x : f(x) > 0 \}$. Show that the set ...
0
votes
1answer
11 views
Distributive function of the sum of two measurable functions.
Let $(X, \mu)$ a measurable space and let $f,g:X \to \mathbb{C}$ complex measurable functions. We define the Distributive function of $f$ by
$$D_f(\lambda) = \mu(\{x : |f(x)| > \lambda \}) $$
I'm ...
0
votes
1answer
60 views
Showing that a certain “norm-like” function fails to satisfy triangle inequality
For any symmetric measurable function $h: I \times I \to \mathbb{R}$, define $$|h| =\sqrt[6]{\int h(x,y)h(x,y')h(x',y)h(x',y')h(x,x')h(y,y') d\mu(x,y,x',y')} $$ where $\mu$ denotes the Lebesgue ...
0
votes
2answers
63 views
Prove that $\mu((0,\infty))=0$.
Suppose $\mu$ is a measure on the Lebesgue measurable subsets of $\mathbb{R}$ and assume that there is a $K\geq 0$ such that for all $n\in\mathbb{N}$, we have $$\displaystyle\int_\mathbb{R} e^{nx}\ d\...
1
vote
2answers
32 views
Continuous functions, null sets and Lebesgue measurable sets
so im trying to prove that if i have a continuous function then f transforms null sets in null sets if and only if f transform Lebesgue measurable sets in Lebesgue measurable sets. Anyone has got some ...
0
votes
1answer
17 views
Find a function sarting with a Lebesgue-Stieltjes measure
Exercise. Find a non-decreasing and right-continuous function $F\colon\Bbb{R}\to\Bbb{R}$ whose Lebesgue-Stieltjes associated $\mu_F$ satisfies the following conditions at same time
$\mu_F(\{0\...
2
votes
2answers
17 views
A collection that is an algebra and countable additivity
Let $X$ be a countable and infinite set and $$\mathcal{F}:=\{A\subset X;\#A<\infty\ \text{or}\ \#A^C<\infty\}.$$
(a) Show that $\mathcal{F}$ is an algebra.
(b) Given $A\in\mathcal{F}$, ...
2
votes
1answer
36 views
Does the Lebesgue measure on the segment $y=x$ represent this distribution?
Set $\Omega=(-1,1)^2$. Consider the following measure on $\Omega$:
$\mu(A)=m(A \cap L)$, where $L=\{ (x,x) \, | \, -1 < x < 1\}$ (the segment of the line $y=x$ in $\Omega$) , and $m$ is the ...
0
votes
2answers
32 views
coin/ sigma-algebra
You flip a coin two times. You consider two events:
$$A=\{ " it \ lands\ heads \ up \ two \ times"\}$$
$$B=\{ " it \ lands\ tails\ up \ two \ times"\}$$
Which events do I have to add to get an ...
0
votes
0answers
64 views
about Lebesgue measure on R
E is a set such that $m(E)>0$, $E \subset (0,1)$ and there exist $c>0$ such that for some moving interval $I$, $$\lim_{mI\rightarrow0}\frac{m(E\cap I)}{m(I)}=c$$
Proof:mE=1
My attempt ,I have ...
2
votes
1answer
33 views
ODE with discontinuous vector field
Consider the ODE
$$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x)), \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$
$$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$
where $\mathbf b = (0,\chi_{\{x_1 \...
2
votes
0answers
28 views
A measure theoretic Lipschitz condition
Let $f$ be a measurable function satisfying following condition:
for every $\epsilon$, we have
\begin{equation*}
\limsup_{\delta \to 0} \bigg\{ \frac{1}{\delta^N} \mathcal L^{2N} \Big( \Big\{ (x,y) \...
1
vote
1answer
44 views
Convergence in probability of running maximum
Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
