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

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4
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1answer
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+100

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]\...
-1
votes
1answer
18 views

Mensuration- Cubes [on hold]

A cube of 64 cubic ft is cut with a plane passing through two diagonally opposite edges.What is the increase in total surface area of the two pieces over that of the original cube?
0
votes
0answers
12 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\...
0
votes
0answers
11 views

What's the flux of a surface pushed by a periodic orbit with Dirac measure?

Let $S$ be some surface in $\mathbb R^3$ and $\phi: \mathbb R \to \mathbb R^3$ a periodic orbit transverse to $S$. Endow $\phi$ with a linear Dirac measure. What's the flux of the surface $S$ pushed ...
0
votes
1answer
33 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 \...
5
votes
0answers
32 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 ...
4
votes
1answer
57 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
1answer
13 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 \...
2
votes
1answer
31 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:...
4
votes
1answer
123 views

Weak Law of Large Numbers

The Weak Law of Large Numbers is often stated with the iid assumption for the underlying RV's. However, I have seen the independence assumption being diluted to the "uncorrelatedness" assumption (e.g.,...
2
votes
2answers
34 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\...
3
votes
1answer
43 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|...
0
votes
1answer
39 views

Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
7
votes
1answer
4k views

Continuity from below and above

In Folland's Real analysis, two of properties of measures are stated as follows: Let $(X,\mathcal{M}, \mu)$ be a measure space. Continuity from below: If $\{E_j\}_1^{\infty} \subset \mathcal{M}$ ...
1
vote
1answer
17 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 ...
4
votes
1answer
41 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 \...
2
votes
1answer
40 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}^...
1
vote
0answers
38 views

$\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ - definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
2
votes
0answers
151 views

Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
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votes
0answers
22 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
1
vote
1answer
29 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}(\...
1
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0answers
47 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
0
votes
1answer
12 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
32 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) = \...
1
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0answers
20 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 ...
0
votes
0answers
19 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 $\nu=0$?
1
vote
1answer
24 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 ...
2
votes
1answer
44 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 ...
1
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0answers
20 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 $y\mapsto \mu(E^y)$ ...
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0answers
20 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\...
-1
votes
1answer
13 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}]...
-2
votes
1answer
18 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}] < \...
0
votes
1answer
16 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?
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votes
0answers
27 views

How to prove that probability for different initial conditions to yield similar trajectory is very small?

For $\epsilon > 0$, suppose $f$ is a function describing chaotic dynamics. Then, for any two different initial conditions, $x,y$, the trajectory obtained is by repeated application of the function $...
1
vote
1answer
51 views

Generating set of Baire sigma-algebra

I got the following statement to prove: Let $X$ locally compact and $\operatorname{Ba}(X)$ the Baire-$\sigma$-algebra, i. e. the smallest $\sigma$-algebra with respect to which all functions in $f \...
3
votes
1answer
9 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
11 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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
3
votes
1answer
26 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^- = \...
4
votes
1answer
54 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(...
1
vote
1answer
24 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
26 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 $\mathcal A$-...
1
vote
1answer
71 views

Using Fatou's Lemma to Prove Monotone Convergence Theorem

Monotone Convergence Theorem- If $\{f_n\}$ is a sequence in $L^{+}$ such that $f_j\leq f_{j+1}$ for all $j$, and $f = \lim_{n\rightarrow \infty}f_n(=\sup_{n}f_n)$, then $\int f = \lim_{n\rightarrow\...
0
votes
1answer
19 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 $\{\{\emptyset\},\{X\}\}$?
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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)\...
2
votes
1answer
48 views

Let $A$ be a Lebesgue measurable set and let $0\leq b\leq\mu(A)$. Show there is measurable $B\subset A$ with $\mu(B)=b$

Let $A$ be a Lebesgue measurable set and let $0\leq b\leq\mu(A)$. Show there is measurable $B\subset A$ with $\mu(B)=b$. This is a qual problem. I tried approaching this with inner regularity. For ...
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 ...