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

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Topology of weak convergence, linear functionals and probabilistic intuition

One very basic question regarding the topology of weak convergence. We know that given the following: $X$ metrizable topological space, $\mathcal{B} (X)$ Borel $\sigma$-algebra, $\Delta (X)$ ...
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12 views

My question is in measure theory in convergence theorems [duplicate]

My question is in measure theory in convergence theorems We know increasing convergence theorem and its proof , its proof is so easy and there is no problem but someone told me you can prove it by ...
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0answers
32 views

A problem from Folland's Real Analysis Book [on hold]

If $E $ is a Borel set in $R^n $,then $D_E (x) =1 $ for almost all $ x \in E $ and $D_E (x)=0 $ for almost all $ x \in E^c $ where $ D_E (x)= \lim_{ r \to 0} m(E \cap B(x,r))/m(B(x,r)) $ This Is my ...
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0answers
5 views

Interpretation of $\sigma$-algebra and filtrations (follow-up question)

This is a follow-up question to Interpretation of sigma algebra, particularly to Jun Deng's excellent answer. He used the example of two coin tosses to explain some fundamentals of how filtrations and ...
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2answers
122 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
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1answer
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Problems on Integration By Parts for Lebesgue Measures

If $F $,$G $ are absolutely continuous function on $[a,b]$, prove that so is $FG $ and $$ \int_{a}^{b} (FG' + GF')(x)dx$$ $= F(b)G(b)-F(a)G(a) $ This is my homework problem.I can't solve this ...
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0answers
22 views

Haar measure on locally compact group

Please I need a help to solve two problems in the book of principles of Harmonic analysis of Deitmar and Echterhoff Exercise 1.4 Let $G$ be a locally compact group with Haar measure $\mu$, and let ...
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0answers
13 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
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1answer
25 views

One question about measure theory [on hold]

Let $(X,S,μ)$ be a finite measure space $μ(X) < \infty $ and $α$ is a finite positive measure on $S$ If $α(A)=\int_{A}{}hdμ$ where $ h \in L^1(μ)$ Prove that $α<<μ$
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0answers
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Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.

Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure. My work: I proved the ...
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1answer
12 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
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1answer
17 views

Calculate the Lebesgue integral of a step function

I'm having some trouble with this problem. Let $$f(x)= \begin{cases} 1 &\text{for}\,\, x = \frac{1}{n}\,,\, n=1,2,\cdots \\ 2 &\text{otherwise} \end{cases}$$ Compute the value of the ...
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1answer
15 views

Independence of sigma algebras of sigma algebras

I have a bunch of questions all of which more or less fall under the subject in the title. The first one goes as follows. Let $E_1,E_2,\ldots,E_n$ be collections of measurable sets on ...
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1answer
18 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
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1answer
73 views

Showing the almost everywhere equality of two unions of sets

Sorry for the ambiguous title. I am trying to prove this seemingly simple statement. Let $\{A_i\}$ and $\{B_i\}$ be sequences of measurable sets in a measure space $(X,\mathcal{A},\mu)$ such that ...
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1answer
39 views

almost sure convergence for non-measurable functions

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume for each $n$, $Y_n:\Omega\rightarrow\mathbb{R}$ is a function but $Y_n$ is not necessarily $\mathscr{F}$-measurable. In this case, is it ...
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0answers
15 views

Do the eigenvectors of a random orthogonal matrix have Haar measure?

For orthogonal $Q$ with Haar measure, does the group of unitary matrices $U$ which diagonalize $Q=U\Lambda U^H$ have Haar measure? I'd be happy to know any answer, even if it's only true for certain ...
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1answer
17 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
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1answer
35 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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0answers
16 views

Polar decomposition of a measure

Prove : If $\nu$ is a complex-valued measure on a measurable space $(X,S)$, then there exists a non-negative real-valued measure $\mu$ on $S$, and a complex-valued measurable function $u$ on $X$ such ...
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1answer
17 views

Countable partition of a probability space

I am trying to prove the following statement. Even though it seems almost obvious that it must be true, I am having trouble with making my arguments precise. Let $\{D_i: i\in \mathbb{N}\}$ be a ...
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0answers
20 views

Trouble to calculate a $\| \;\; \|_{L^1}$ norm ($\int\limits_0^1 n e^{-nx}dx$)

Let $I = (0, 1)$ and $f_n(x) = n e^{-nx}$ a sequence of functions. I calculated $\|f_n\|_1$ and I obtained two different aswers. $$\|f_n\|_1 = \int\limits_0^1 |f_n(x)| dx = \int\limits_0^1 n e^{-nx} ...
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1answer
15 views

measurable function and properties of their integrals [on hold]

I have to prove that given $(X,M,\mu)$ a measurable space and two measurable functions $\omega_i\colon M \rightarrow [0,\infty]$, $i=1$, $2$, if for all measurable sets E holds $\int_{E} \omega_1 = ...
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1answer
20 views

Measure of open sets covering compact set

Prove that if $F$ is a finite collection of open intervals that covers a compact interval $[a, b]$, then the sum of the lengths of the intervals in the collection is strictly greater than $b − a$ ...
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1answer
31 views

Components of $\sigma$ algebras

I previously thought that $\mathcal{B}(\mathbb{R})^n = \mathcal{B}(\mathbb{R}^n)$, which I just realized was false! I'm wondering whether the following weaker statements are true: Let $\mathcal{F}$ ...
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2answers
40 views

Rudin's proof of Riesz representation theorem

Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the set of complex valued continuous functions of compact suppot on $X$, $C_c^+(X)$ the set of non-negative real valued functions in ...
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0answers
35 views

Sum of not necessarily independent discrete random variables.

Let ${X_k}$ be a sequence of discrete random variables, where $P(X_k=k)=\frac{1}{k^2}$ and $P(X_k=0)=1-\frac{1}{k^2}$. Let $S_n=\sum_{i=1}^n X_i$. Does $\frac{S_n}{\sqrt{\log n}}\rightarrow 0$ (in any ...
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0answers
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How to compare the Hardy-Littlewood maximal function for balls and cubes?

I am currently working through a set of notes I found on the internet at: http://math.msu.edu/~charlesb/Notes/DuoChapter2.pdf I am up to page 8, and the Hardy-Littlewood maximal function for balls ...
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26 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
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1answer
28 views

Is there a measure invariant with respect to the Möbius transformation?

I would like to use a measure ${\rm d} \mu (z)$ on ${\mathbb C}$ so that for any $f(z)$ $$\int_{\mathbb C} f(z) {\rm d} \mu (z)$$ is invariant under Möbius transformations. Taking the ...
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What does it mean “rotationally invariant density”?

In the great answer given by the math.SE user @Tim, he does 2 hypothesis, on of the which ones is about the rotationally invariance of the density. Can you explain formally what does it mean? I do ...
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2answers
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Mean value theorem for sliding window of Lebesgue integral of integrable function

Take $f \in L^1(\mathbb{R})$ and define $g(x) = \int_x^{x+1} f(t) \, dt$. If $g(a) > 0$ and $g(b) < 0$, is it necessarily true that there is some $c \in [a,b]$ such that $g(c) = 0$? I feel as ...
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2answers
36 views

$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable

I know there are similar questions up proving this, but I had a question specific to the following proof (specifically in bold): Let $f$ be a Lebesgue measurable function on $\mathbb{R^n}$. Then the ...
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2answers
74 views

If a function is Lebesgue measureable, does this imply Lebesgue integrability?

Say we take the measure of a countable set, we obtain that $\mu=0$. Now if this is the case, does this automatically imply that it is Lebesgue integrable as well? The reason I bring up the set is ...
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2answers
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a interesting question from topological group

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$. My question is: Can ...
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1answer
37 views

Show that there exists a measurable function $g$ with $|g| \leq 1$ so that $\int_E g d\nu = |\nu|(E)$ for all measurable sets $E$.

Let $(\mathcal{X},\mathcal{M},\nu)$ a measure space where $\nu$ a signed measure. Show that there exists a measurable function $g$ with $|g| \leq 1$ so that $\int_E g d\nu = |\nu|(E)$ for all ...
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1answer
39 views

On the measurability of a special set

Is $\underset{n=1}{\overset{\infty}{\cup}}[-n,n]$ measurable? This is a question on Measure Theory of a previous exam period. I can't understand if it's that easy as I find it, or I'm missing ...
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1answer
21 views

Coniditional expectation for bounded random variables

I need to show that for bounded random variables $X$ and $Y$ we have ${\rm{E}}[X{\rm{E}}(Y||\mathcal{F})]={\rm{E}}[Y{\rm{E}}(X||\mathcal{F})]$. The only property of conditional expectation I am aware ...
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2answers
33 views

F measurable and conditional expectation.

(a):I found it easily cause sum of measurable sets are measurable. (b),(c): I know limsup(Sn/n) is also measurable but I can't prove that just sup(Sn/n) is measurable. (d): I solved it by using the ...
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1answer
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Complete separable metric space X represented represented as union of closed sets

I have a problem concerning a statement I found in volume 2 of the classic reference book on measure theory by Bogachev. More precisely, I have a problem concerning theorem 6.1.13. I the proof the ...
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1answer
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Measure space $(X,\mathcal{F},\mu)$ where $L^p(X,\mathcal{F},\mu) \neq L^q(X,\mathcal{F},\mu)$ if $p\neq q$

I was trying to solve this problem: Let $(X,\mathcal{F},\mu)$ be a measure space where $L^p(X,\mathcal{F},\mu) \neq L^q(X,\mathcal{F},\mu)$ when $p\neq q$. Prove that there exist a sequence of sets ...
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Density of simple functions in $L^\infty$

$\textbf{Theorem}$ Is the set of simple measurable functions $s$ (with values in $\mathbb{R}$ or $\mathbb{C}$) such that $ \mu(\{x: s(x)\neq 0\}) <\infty$ is dense in $ ...
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1answer
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Why do we use non-negative functions in measure theory

I've just started learning Measure theory and I was curious if there is a rationale for working with non-negative functions as the 'base' upon which more complex theorems are built. Why not include ...
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Lebesgue integral over subset

I have a basic question related to Lebesgue integration restricted to a subset of $\mathbb R^n$. In general, for $(X,\Sigma,\mu)$ a measurable space, if $f$ is measurable one can calculate the ...
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1answer
32 views

Measurable set as a subset of a non-measurable set

is it possible to find a measurable subset $A$ of a non-measurable set $B$ such that $A$ can be assigned the measure zero or more particularly a non zero value ( in the case of the Lebesgue measure ...
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18 views

A property on exterior measures on a metric space

Let $(X,d)$ be a metric space and $\mu:2^X\to[0,\infty]$ be an exterior measure on $X$, $\mathcal{M}:=\{E\subset X:\forall S\subset X\ \ \ \mu(S)=\mu(S\cap E)+\mu(S-E)\}$. Suppose ...
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0answers
26 views

Kolmogrov 0-1 law

Let {$A_n$} be an independent sequence of events. Show that the event $$(\omega:\frac{\sum_{k=1}^n I_{A_k}(\omega)}{n} \rightarrow x)$$ has probability either 0 or 1. I think if I show this event as ...
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2answers
44 views

Non-measurable subsets of a null set

Given the Lebesgue measure on $\mathbb{R}$, I am wondering if a non-measurable set can always be included in a null set? More precisely, let $A$ be a set in the Borel σ-algebra ...
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33 views

Inner regularity of the Haar measure

Let $G$ be a locally compact Hausdorff topological group, and denote by $B$ the σ-algebra generated by the open subsets of $G$, an element of $B$ is called a Borel set. A (left) Haar measure m ...
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1answer
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Kolmogorov 0-1 law, Measure Theory

Suppose that $(X_1, X_2,...)$ is an independent sequence of random variables and $Y$ is measurable $\sigma(X_n,X_{n+1},....)$ for each $n$. Show that there is a constant a such that $P(Y = a) = 1$. I ...