Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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4
votes
0answers
35 views

Is $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$ a $\sigma$-algebra of subsets of $X$ or not? [closed]

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define$$\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}.$$Is $\mathcal{B}$ a $\...
2
votes
1answer
19 views

Ascending chain of monotone classes, $A$ necessarily in $\mathcal{M}$

Suppose $\mathcal{M}_1 \subset \mathcal{M}_2 \subset \ldots$ are monotone classes. Let $\mathcal{M} = \bigcup_{n = 1}^\infty \mathcal{M}_n$. Suppose $A_j \uparrow A$ and each $A_j \in \mathcal{M}$. Is ...
2
votes
2answers
37 views

Real Analysis, Folland Proposition 2.30 Modes of Convergence

Proposition 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{f_{n_j}\}$ that converges ...
2
votes
0answers
26 views

If $\mu(E)\geqslant 0$ is it true that $E\in \mathfrak{M}$?

Suppose $(X,\mathfrak{M},\mu)$ be a mesure space. Let $E$ such that $\mu(E)\geqslant 0$. Can we conclude that $E\in \mathfrak{M}$? I think YES because $\mu$ is the set function with domain $\mathfrak{...
1
vote
2answers
23 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
6
votes
1answer
96 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
2
votes
0answers
47 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
1
vote
0answers
16 views

Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
1
vote
0answers
58 views

Construction of Lebesgue measure in Rudin's RCA book

This theorem from Rudin's RCA book. Here's one moment from it's proof which seems to me very weird. Rudin states that equality $\lambda(E)=m(E)$ holds for all Borel sets. But I think that it's ...
0
votes
1answer
36 views

Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
0
votes
0answers
95 views

convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
2
votes
2answers
46 views

Real Analysis, Folland Proposition 2.29 Modes of Convergence

Background Information: $f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$ A sequence $\{f_n\}$ of measurable complex-...
1
vote
1answer
59 views

If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
1
vote
0answers
16 views

The finite-dimensional distribution of a stochastic process

Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties: $K$ is symmetric ($K(s,t)=K(t,s)$). $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\...
6
votes
0answers
65 views

Kolmogoroff 0-1 does this proof work?

I have thought at this proof of the Kolmogorov 0-1 Law varying a little the sketch found in Probability essentials (Jean Jacod, Philip Protter). My questions are Is it a valid proof? Is it a bad ...
0
votes
1answer
17 views

Infinities on null sets

This is a conceptual question! Why is it that (e.g.) $\int_0^1 \frac{1}{x} dx$ doesn't converge. I'm stuck in the following way of thinking about it: Since the problematic part is $\int_0^\epsilon \...
3
votes
2answers
45 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
1
vote
1answer
24 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
2
votes
1answer
47 views

Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
3
votes
1answer
49 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...
1
vote
1answer
41 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
0
votes
1answer
14 views

Ash and Doleans-Dade Probability and Measure Theory Section 1.2 Question 2

Ok so in section 1.2 of chapter 1, the authors pose the following challenge: Let $\mu$ be the counting measure on $\Omega$, where $\Omega$ is an infinite set. Show that there is a sequence of sets $...
0
votes
1answer
35 views

Real Analysis, Folland Problem 2.3.19 Integration of Complex Functions

Problem 2.3.19 - Suppose $\{f_n\}\subset L^1(\mu)$ and $f_n\rightarrow f$ uniformly. a.) If $\mu(X) < \infty$, then $f\in L^1(\mu)$ and $\int f_n \rightarrow \int f$. b.) If $\mu(X) = \...
0
votes
0answers
21 views

Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
4
votes
1answer
52 views

Conditions on a complex measure to be real

Let $(X,\mathcal{S}, \mu)$ be a measure space with $X$ a locally compact Hausdorff space, $\mathcal{S}$ the Borel subsets of $X$ and $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \...
1
vote
0answers
23 views

Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
-2
votes
1answer
32 views

Uniform Integrability - different characterisation - prove (ii)

Probability with Martingales: For the 'only if' part assuming the hint is true, then I guess we have $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \...
-1
votes
1answer
19 views

Uniform Integrability - different characterisation - prove hint

Probability with Martingales: For the 'only if' part how to prove the hint? i'm guessing it's something to do with $$E[X 1_F] \le E[X1_{\Omega}]$$ $$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]...
0
votes
1answer
21 views

Uniform Integrability - sufficient condition and bounded convergence theorem with weaker hypothesis

Probability with Martingales: How does the result follow? Do we choose $K = (\frac{\varepsilon}{A})^{\frac{1}{1-p}}1_{A \ne 0}$ Why do we have that inequality?
1
vote
0answers
23 views

Alternative Proof about Product Measures and Iterated Integrals

Background Theorem 2.36 of Folland's Real Analysis says that if $(X,M,\mu)$ and $(Y,N,\nu)$ are sigma finite measure spaces, and $E\in M\bigotimes N$, then $x\mapsto \nu(E_x)$ and $y\mapsto \mu(E^y)$ ...
3
votes
1answer
14 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
0
votes
1answer
43 views

Real Analysis, Folland Theorem 2.26 Integration of Complex Functions

Background information: Theorem 2.10 - Let $(X,M)$ be a measurable space. a.) If $f:X\rightarrow [0,\infty]$ is measurable, there is a sequence $\{\phi_n\}$ of simple functions such that $0 \...
2
votes
1answer
41 views

Real Analysis, Folland Theorem 2.25 Integration of Complex Functions

Theorem 2.25 - Suppose that $\{f_j\}$ is a sequence in $L^1$ such that $\sum_{1}^{\infty}\int |f_j| < \infty$. Then $\sum_{1}^{\infty}f_j$ converges a.e. to a function in $L^1$, and $$\int \sum_{1}^...
2
votes
1answer
50 views

Real Analysis, Folland The Dominated Convergence Theorem

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Question: 2.24 The Dominated Convergence Theorem - Let $\{f_n\}$ be a sequence in $L^1$ such that ...
3
votes
1answer
28 views

Real Analysis, Folland Proposition 2.22 Integration of Complex Functions

Proposition 2.22 - If $f\in L^1$, then $|\int f|\leq \int |f|$ Attempted proof - If $f$ if a real-valued function then $$\left|\int f\right| = \left|\int f^+ - f^-\right|\leq \int f^+ + \int f^- = \...
1
vote
1answer
26 views

Real Analysis, Folland Proposition 2.21 Integration of Complex Functions

Proposition 2.21 - The set of integrable real-valued functions on $X$ is a real vector space, and the integral is a linear functional on it. Attempted proof - Note that we can derive the axioms of a ...
0
votes
1answer
14 views

Derivative of volume of given set

As picture below ,how to compute the $\partial_t |\Omega_t|$ ? The picture below is from the 32 page of Maximum principles and the method of moving planes. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
1
vote
1answer
25 views

Can anyone explain one trivial theorem about measurability of functions?

I know this is simple, but I just cannot get my head around. Can anyone explain the following? if $\mathcal{C} \subset\mathcal{B}$ and $\sigma(\mathcal{C})=\mathcal{B}$, then $h^{-1}:\mathcal{C}...
3
votes
0answers
36 views

If $f_n \to f$ and $g_n \to g$ in measure and $\mu$ is finite, then $f_n g_n \to fg$ in measure

This is Problem 3.1.5 in Cohn's Measure Theory, 2nd edition. Let $\mu$ be a measure on $(X, \mathcal A)$, and let $f, f_1,f_2, \ldots$ and $g,g_1,g_2,\ldots$ be real-valued $\mathcal A$-...
0
votes
1answer
20 views

Empty set in an algebra or sigma-algebra

Does an algebra (or a sigma-algebra) contains the empty set or a set containing the empty set? E.g., let $X$ be a set. Is the trivial sigma-algebra $\{\emptyset,X\}$ or $\{\{\emptyset\},\{X\}\}$?
4
votes
1answer
65 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
0
votes
0answers
22 views

On bounding $L^p$ norms

Jensens' inequality states that if $f\in L^1$ and $\varphi$ is convex, then $$\require{esint}\varphi\left(\diagup\hspace{-11pt}\int_xf\,\mathrm{d}x\right)\le\diagup\hspace{-11pt}\int_X(\varphi\circ f)\...
1
vote
0answers
26 views

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)?

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)? This is from the last sentence in the proof in the following ...
5
votes
1answer
123 views

Diffuse-like decomposition of the segment $[0,1]$ in accordance with Lebesgue measure

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\...
2
votes
0answers
5 views

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...
1
vote
2answers
27 views

For any measurable set $A\subset\mathbb{R}$ and $r\in(0,\mu(A))$ we have $(\mu|_{2^A})^{-1}(r)\neq\emptyset$

Recently when I tried to prove a statement I needed to rely on the following fact that intuitively feels correct, but I wasn't able to prove it accurately. Here it is: Consider a set $A\subset\...
-1
votes
0answers
21 views

Why is the bounded linear functional $T(g)=\int_X fg d\mu$ an isometry?

Royden claims the following in Real Analysis on page 400. T : $L^q$(X, μ) -> ($L^p$(X, μ))* is an isometry. Can anyone explain why is T a mapping from $L^q$(X, μ) to ($L^p$(X, μ))* instead of to ...
8
votes
1answer
80 views

Is there a measure space $(X,\mathcal M, m)$ such that $\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$?

I have in mind the following question: Is there a measure space $(X,\mathcal M, m)$ such that the range of $m$ satisfies $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
1
vote
0answers
20 views

Question on product measure: $\int_{[0,R]^2} g(x,y) df_1(x) \times df_2(y) = \int_{[0,R]} \left( \int_{[0,R]} g(x,y) df_1(x) \right) df_2(y)$ holds?

Suppose I have a real valued positive increasing functions $f_1(x), f_2(y)$. Then we know we can define Riemann-Stieltjes integral by defining measures $df_1(x)$ and $df_2(y)$. Let $g(x,y)$ be a ...
4
votes
1answer
58 views

Measurable function and the Mean Value Theorem

Let $\,f:[a,b]\to \mathbb{R}\,$ be continuous on $[a,b]$ and derivable on $(a,b)$. By the mean value property, for all $\,x\in (a,b)\,$ there exists $\,\xi_x\in (a,x)\,$ such that $\,f(x)-f(a)=f'\left(...