Questions relating to measures, measure spaces, Lebesgue integration and the like.
2
votes
2answers
23 views
Baire's theorem from a point of view of measure theory
According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it ...
0
votes
0answers
25 views
Given $e \in L(0,1)$, why is $\gamma(t)=\int_0^1 G(t,s) e(s)\, ds$ in $C^2[0,1]\cap C^3(0,1)$?
If I have a function $e \in L(0,1)$, why is the function $\gamma$ defined by $$\gamma(t)=\int_0^1 G(t,s) e(s)\, ds$$
an element of $C^2[0,1]\cap C^3(0,1)$, where:
$G$ is the Green's function of BVP ...
0
votes
0answers
16 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
1
vote
0answers
49 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
vote
1answer
18 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\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
0answers
24 views
Measurability of multifunction
Let $f:[a,b]\times \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$. Suppose $ f (.,x, u) $ is Lebesgue measurable for each $(x,u)$. Suppose also that $ f $ is continuous at $ (x, u) $ ...
15
votes
5answers
252 views
Examples of properties that hold almost everywhere, but that explicit examples unknown.
In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero.
I think it is very ...
0
votes
2answers
23 views
Two random variable with the same variance and mean
Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely.
My partial answer:
By the definition of conditional expectation we have ...
2
votes
0answers
33 views
Alternative rigorous definition of a surface integral
Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
3
votes
1answer
41 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
3
votes
1answer
57 views
Measures on all subsets of $\aleph_0$
A theorem of Ulam says:
A finite measure $\mu$ defined on all subsets of a set of cardinality $\aleph_1$ must be $0$ for all subsets if it sends every $1$-element subset to $0$.
Will this ...
1
vote
0answers
20 views
Lebesgue measure of set $M = \{ [x,y] \in \mathbb{R}^2; 2 < x + y < 3; x < y < 3x \}$?
although we can do this by splitting the area four ways and computing four integrals, my book suggests that I try the substitution $ u = x + y$ and $ v = \frac{y}{x}$.
I expressed $x$ and $y$ in ...
0
votes
0answers
33 views
Proving that $\bigotimes_{i=1}^n \cal{B}_{X_i} = \cal{B}_{X}$
Theorem: Given separable metric spaces $X_1,\ldots,X_n$ and $X=\prod_{i=1}^n X_i$, where $X$ has the product metric $d(f,g)=\sqrt{d_1 (f(1),g(1))^2 +\cdots + d_n (f(n),g(n))^2}$. Then ...
1
vote
1answer
25 views
Inequality between 2p-norm and p-norm for random variables
Recently I was studying something about random matrix theory, and class of sub-guassian / sub-exponential random variables is of interest. In the literature it gave an inequality as following:
...
3
votes
1answer
44 views
The semifinite portion of a measure $\mu$
Let $\mu$ be a measure and define $\mu_1$ such that $\mu(E)=\mu_1(E)$ for $\mu(E)$ finite. And for $\mu(E)$ infinite definite $\mu_1$ such that:
(i) if $E$ contains finite subsets of arbitrarily ...
2
votes
4answers
87 views
Book Suggestions for an Introduction to Measure Theory [duplicate]
Couldn't find this question asked anywhere on the site, so here it is! Do you guys have any recommendations for someone being introduced to measure theory and lebesgue integrals?
A mentor has ...
0
votes
1answer
28 views
A Measure For The Space of Probability Density Functions
Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
4
votes
1answer
45 views
Measurability of an Indexed Product-Measure
If for any fixed $\omega_1$, $P_{\omega_1}$ is a probability measure and $Q_{\omega_1}$ is a stochastic kernel and both are measurable in $\omega_1$, is the indexed product measure ...
5
votes
3answers
152 views
why measure theory
I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary ...
1
vote
1answer
23 views
Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in ...
1
vote
1answer
44 views
Isomorphism Subalgebra
Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
2
votes
1answer
48 views
Simplification of an expression
How do I simplify the following expression?
$$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$
where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e ...
3
votes
1answer
28 views
Are the continuous functions on $G$ dense in $L^{1}(G)$?
If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
2
votes
1answer
32 views
Basic question about the definition of an integral on a measure space
Let $(X,\mathcal{B},\mu)$ be a measure space.
$\bf{\text{Definition:}}$ For a non-negative measurable function $f$ on $X$, $E\in \mathcal{B}$,
$$\int_{E}f d\mu := \text{inf}\int_{E}\varphi d\mu$$
...
3
votes
1answer
77 views
If $f$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions such that $s_n \longrightarrow f \; a.e$?
If $f:[0,1]\longrightarrow\mathbb{R}$ is a bounded measurable function $\Longrightarrow$ there is a sequence of step functions $\displaystyle s_n=\sum_{j=1}^{p} c_j \cdot \chi _{I_j}$ such that $s_n ...
0
votes
1answer
21 views
Conditional expectation is square-integrable
I am given the following definition:
Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events ...
0
votes
1answer
32 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
0answers
60 views
Let $g$ be a bounded measurable function on $[0,1]$.
Let $g$ be a bounded measurable function on $[0,1]$.
For each $n$
Let $\displaystyle I_j=j\cdot \frac{1}{2^n}+[0,\frac{1}{2^n}] $ , $j=0,1\cdots ,2^n-1$ , a partition of $[0,1]$ by bisections
...
3
votes
1answer
38 views
A two-dimensional set of measure zero
I have a 2D domain $[0,1]\times[0,1]$. This domain contains some set of measure zero $A$, the last understood as the Lebesgue measure in $\mathbb{R}^{2}$.
Is the following true: for almost all ...
2
votes
1answer
67 views
How to understand C(X)'' = bounded Borel measurable functions?
Let $X$ be a compact metric space and $C(X)=\{ f:X\rightarrow \mathbb{R} \ | \ \ f \ continuous\}$ with the uniform norm. It is a separable Banach space.
1) I'm aware of the fact that $C(X)^*$, the ...
2
votes
1answer
48 views
Riemann integral with intervals?
Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$
Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$.
You can use the definition of Riemann integral ...
1
vote
1answer
35 views
Counterexample to upper continuity
Let $M$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu:M\rightarrow[0,\infty)$ be a finitely additive set function. I'm trying to decide if it's automatically true that for all ascending ...
4
votes
0answers
38 views
Which definition is correct?
I have encountered several different definitions of left Haar measure that don't all seem to agree.
The setting I care about is Locally Compact Groups.
The first seems to completely disagree with ...
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
35 views
Haar measure $\tau$-additive?
I'm reading some results from Measure Theory Volume 4 by D.H. Fremlin, and I'm stuck on something.
This is pulled out of one of his lemmas (stated more generally for topological groups):
A Haar ...
0
votes
1answer
74 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 ...
-1
votes
0answers
24 views
Lebesgue inner measure
the definition of inner measure: m ∗ (A)=sup{m(S):S∈M,S⊆A} I need to prove:
1) If inner measure=outer measure then A is measurable set
2) m*(AUC)+m*(A n C)>m*(A)+ m* (C)
3)m*(UA)> sum (m*(A)) for ...
3
votes
1answer
32 views
Why are Haar measures finite on compact sets?
I'm working through the answer by t.b. to another user's question here:
A net version of dominated convergence?
because I am trying to work through a related problem and I think it will be ...
3
votes
0answers
20 views
Are Haar measures complete?
If $G$ is a locally compact group and $\mu$ is a left Haar measure for $G$, then is the measure space $(G,B(G),\mu)$ complete (where $B(G)$ is the set of Borel subsets of $G$)?
Or do we have to take ...
1
vote
0answers
44 views
Proving that $ \int \left| f-g \right|~d\mu = 2\int_{A_0} (f-g)~d\mu$
Given a (dominant) measure $\mu$, consider two probability measures $f~d\mu$ and $g~d\mu$ over $(\Omega, \mathcal F)$, I'd like to check the following reasoning for showing that
$$ \int \left| f-g ...
1
vote
1answer
24 views
Showing that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}[\exp[tX]]$
The problem is to show that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}(\exp[tX])$ given $\exp(tX)<\infty$ for $t\in \mathbb{R}$ where $X$ is a random variable.
Then to show that ...
5
votes
1answer
39 views
Finitely additive measure on $\mathbb R$
Suppose $\mathcal B$ is the Borel $\sigma$-algebra on $\mathbb R$. Let $\mu : \mathcal B \rightarrow [0, \infty ]$ be a finitely additive(but not necessarily countably additive), ...
1
vote
0answers
15 views
Convergence of an Integral in a locally compact group
I'm trying to finish an exercise which I asked about earlier here:
Mapping $G$ into its group algebra as left multiplication. Continuous?
$\bf{\text{The setting:}}$
Let $G$ be a locally compact ...
0
votes
0answers
31 views
question about essential supremum
Consider $u : \Omega \rightarrow \mathbb{R}$ a measurable, nonnegative and bounded function. With $\Omega \subset \mathbb{R}^n$ bounded and open .
Is true that $\mathrm{ess } \inf \ u = \inf \ ...
0
votes
1answer
29 views
$\int_\Omega fd\mu_n\to\int_\Omega fd\mu,\ \forall\ f\in C_0(\Omega)$ implies $\mu_n(\Omega)\to \mu(\Omega)$?
Let $\Omega\subset\mathbb{R}^N$ be a bounded open smooth domain and $C_0(\Omega)$ the set of bounded continuous functions with compact support. It is know that $C_0(\Omega)^\star =M(\Omega)$, where ...
2
votes
0answers
25 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, ...
2
votes
1answer
36 views
Calculate an integral in a measurable space
Let $(X,\mathcal{M})$ a measurable set with measure $\mu$. Let $f$ be an integrable non negative function, such that $K:=\int_{E}f \mathrm d\mu<\infty$, where $E\in(X,\mathcal M)$. Let ...
2
votes
1answer
58 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
2
votes
0answers
35 views
A few questions about Measure Algebras
I've written up some of my understanding as well as I can of the Measure Algebra, trying to see the details behind a very brief treatment. There a couple places where I cannot make see how to make ...
4
votes
1answer
67 views
Involution in $L^{1}(G)$ is isometric.
(Sorry for asking so many questions of the same type. There is an underlying issue that I think once resolved will allow me to understand them all at once.)
Let $G$ be a locally compact group, and ...
