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

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convergence in distribution in Banach spaces

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
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Measurable isomorphism between two non-totally ergodic systems

Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
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1answer
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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-...
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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 ...
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Prove $X_n \nrightarrow X = \bigcup_{k=1}^{\infty} \{|X_n - X| > \frac{1}{k}\}$

Probability with Martingales: Important inequalities: 1, 2 $$\liminf x_n > z \to \liminf(x_n > z)$$ $$\liminf x_n < z \to \limsup(x_n < z)$$ What I tried: I think the ...
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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\...
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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 ...
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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 \...
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2answers
38 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\...
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1answer
22 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 ...
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1answer
34 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:...
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1answer
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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|...
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1answer
30 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}(\...
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1answer
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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 $...
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1answer
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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) = \...
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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 $\nu=0$?
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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 \...
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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\...
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1answer
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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}] < \...
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1answer
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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}]...
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1answer
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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?
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1answer
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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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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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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 ...
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1answer
37 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 \...
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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}^...
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1answer
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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 ...
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1answer
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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^- = \...
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1answer
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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 ...
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1answer
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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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
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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}...
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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 $\mathcal A$-...
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1answer
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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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1answer
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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 $\...
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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 $...
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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)\...
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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 ...
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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]\...
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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 ...
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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\...
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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 ...
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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 $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
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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 ...
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1answer
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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(...
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2answers
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Theorem 2.17 from RCA Rudin

I understood the proof of points $(a)$ and $(c)$. But I can't understand the proof of $(b)$. It's obvious that every closed set is $\sigma$-compact. But how Rudin applies $(a)$ here? We have to show ...