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

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2
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0answers
551 views

Inner and outer Lebesgue measure

For any subset $E\subseteq I=[0,1]$ define the inner Lebesgue measure by $m_{*}(E):=1-m^*(I\setminus E)$, where $m^*$ is the outer Lebesgue measure. Show that $$ E\subseteq I \mbox{ is ...
2
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1answer
113 views

Calculating the probability of following event involving Brownian motion

I have a big time trouble in evaluating the following probability. It is related to brownian motion and measure, so I am asking experts from both fields for help! Denote $B_t$, $t\in [0, T]$ be ...
9
votes
3answers
276 views

Existence of a Minimal Cover

I'm well aware that for the sequence $x_n=\frac{1}{n}$, $\text{inf }x_n=0$ but $0 \notin (x_n)$. This made me think about something similar but when we are no longer thinking about existence of a ...
1
vote
0answers
120 views

Example of a function that has the Luzin $n$-property and is not absolutly continuous.

The Banach–Zaretsky theorem (page 196) says that a continuous function $f:[a,b]\to\mathbb{R}$ of bounded variation is absolutely continuous if and only if $$E\subset I \text{ has zero Lebesgue ...
5
votes
1answer
238 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
1answer
198 views

Is there any absolutely continuous function $f$ and a null set $X$ such that $f(X)$ is not a null set?

Here (wikipedia) there are some properties of absolutely continous functions. Some of them requires a closed interval to be the domain of $f$. So, I would like an example of an absolutly continuous ...
0
votes
1answer
61 views

Approximating linearly independent functions with linearly independent functions.

Let $(\Omega,\Sigma,\mu)$ be a measure space. Let $f_{1}, ... , f_{n}$ be an $\bf{\text{Auerbach basis}}$ for a finite dimensional subspace $N\subset L_{1} := L_{1}(\Omega,\Sigma,\mu)$. That is, ...
1
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1answer
46 views

Any hint for this measure theory problem?

I need a hint because I really don't know where to start: Let $O \subset \Bbb R$ be open and let $f:O \to \Bbb R$ be a $\mathcal{C}^1$ function. Show that if $A \subset O$ and $m(A)=0$ then ...
1
vote
1answer
59 views

Let $S$ be a subset of $\Bbb R$ of finite Lebesgue measure.

I have to show that there is a $r \in \Bbb R$ such that the Lebesgue measure $m(S \cap (-\infty, r))=1/2 \cdot m(S)$. I've thought in some particular cases: Since $\Bbb Q$ has measure zero, any $r$ ...
3
votes
1answer
593 views

Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?

$m$ is the Lebesgue measure. I was thinking that: $\Bbb Q$ is dense and $m(\Bbb Q)=0<1$ but it fails to be open, but maybe I could construct an open set from this fact, or also using that $(0,1)$ ...
2
votes
2answers
39 views

Proving the equality $]a,b]=\cap_{n \ge 1} ]a, b+\frac 1 n[$

I need some help proving the following simple equality : $]a,b]=\bigcap_{n \ge 1} ]a, b+\frac 1 n[$. $\subset$ is obvious as $\forall n$ we have $]a,b] \subset ]a, b+\frac 1 n[$. But how does one ...
0
votes
1answer
225 views

Uniqueness of elementary measure.

If we define finite union of boxes as elementary sets. Then define a measure of these sets as follows: Let $E \subset \mathbf{R}^d$,If $E$ is partitioned as the finite union $B_1 \cup \cdots \cup ...
2
votes
1answer
154 views

Limit of measure of finite unions

I am beginning to learn measure theory, and I have a basic doubt regarding to measure of union of sets, and limits. Let $\mathcal{A}$ be a $\sigma$-algebra on $X$, and $\mu$ be a finite measure on ...
0
votes
1answer
50 views

Expectation and variance of random series

Consider the random series $1\pm\dfrac12\pm\dfrac14\pm\dfrac18\pm\ldots$ with the assignment of a plus or minus in the $n$th term being decided by the toss of a coin. Compute its expectation value ...
3
votes
2answers
83 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
2
votes
0answers
192 views

Set of points where sequence converges is measurable

Let $f_n:X\rightarrow\mathbb{R}, n=1,2,\ldots$ be a sequence of measurable functions. Show that the set of points $x\in X$, where the sequence $\{f_n(x); n=1,2,\ldots\}$ converges, is measurable. ...
1
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1answer
215 views

Let E be a measureable subset of [0,1] with positive measure. Let $E-E= \{ x-y | x,y \in E\}$Show that this set contains a neighborhood of 0. [duplicate]

Let E be a measureable subset of [0,1] with positive measure. Let $E-E= \{ x-y | x,y \in E\}$Show that this set contains a neighborhood of 0. (hint use inner approximations) I think : this set is ...
4
votes
0answers
136 views

Measurable and coordinate functions in terms of Borel sets

Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a measure on $F$. A map $F=(f_1,\ldots,f_n)$ of $X$ into $\mathbb{R}^n$ is said to be measurable if each of the coordinate ...
3
votes
1answer
38 views

Independent sets in subfield

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $\mathcal F$. Suppose that $A_1,\ldots,A_n$ are independent sets belonging to $\mathcal F$. Let ...
1
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1answer
79 views

New Outer Measure Question

Let $m^*(A)$ be the outer measure of a set $A$ $\in$ $\mathbb{R}$. This is defined by $m^∗(A)$= $inf(Z_A)$ where $Z_A$={ $\sum_{n=1}^\infty l(I_n)$ : $I_n$ are intervals, $A\subseteq$ ...
3
votes
2answers
32 views

Expectation of the squared error with regards to a sub sigma field

I am totally stuck. Given a probability space $(\Omega, \mathcal F, \mathbf P)$ and a random variable $X$. Let $\mathcal A$ be a sub-$\sigma$-field of $\mathcal F$. Let $Y$ run over all $\mathcal ...
1
vote
2answers
145 views

Conditional expectation as expectation

$X$ is a real-valued random variable. $B$ is an event. Is it right that the conditional expectation $E[X\mid B]$ can be seen simply as an ordinary expectation over a new probability space having as a ...
1
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2answers
40 views

discretization of probability measures

Suppose I have given a probability measure $\nu$ over the positive reals. For a fixed $n\in\mathbb{N}$, we set $\lambda := \frac{1}{n}$ and $A_n:=\{\lambda k, k=0,\dots\}$. Now we look at a certain ...
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1answer
76 views

prove that if M*(E) is finite if E is finite then M* is an outer measure

define M*(E) as the number of points in E if E is finite and M*(E)=infinity if E is infinite. Show that M* is an outer measure. determine the measurable sets.??
7
votes
5answers
987 views

Measure theory and topology books that have solution manuals

I am trying to find a book to learn measure theory that contains complete solutions manual. Does someone know of any? Also, I would like to know if there is a book with solutions manuals about ...
2
votes
1answer
65 views

You only need countable many sets each time in a generated $\sigma$-algebra, right?

Let $X$ be a non-empty set, then any $\mathcal{E}\subset\mathcal{P}(X)$ can generate a $\sigma$-algebra $\sigma(\mathcal{E})$. Here, $\sigma(\mathcal{E})$ is the intersection of all the ...
0
votes
2answers
42 views

“A measure whose domain includes all subsets of null sets is called complete”

I took this sentence from Folland's Real Analysis book. However, the "domain" in the sentence looks pretty vague to me. Can someone help me formulate this statement into precise math language? This ...
2
votes
0answers
102 views

Textbook Recommendation; Proability Theory with Measure Theory

I'm currently taking a course in Probability Theory and was hoping someone could point me in the direction of a useful supplementary textbook. Our course currently uses A Modern Approach to ...
2
votes
1answer
29 views

Discrete measure theoretic model for Bernoulli sequence

Let $X=\{x_1,x_2,\ldots\}$ be a countable set, $P_1,P_2,\ldots$ a sequence of non-negative numbers such that $\sum P_i=1$, and $\mu$ the measure $\mu(A)=\sum_{x_i=A}P_i$. Show that $X$ cannot ...
4
votes
1answer
578 views

Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ ...
2
votes
2answers
81 views

Show that $\sigma(\mathscr{C}) = \sigma(\mathscr{G} \cup\mathscr{H})$

I'm doing some grad work as a past-time and I wanted to learn measure theory but I'm absolutely not managing it. Let $\mathcal{G}$ and $\mathcal{H}$ be two $\sigma$-algebras on $\Omega$. Let ...
1
vote
1answer
389 views

Liminf and Limsup in measure theory and in sequences

In measure theory, given sets $A_1,A_2,\ldots$, we define $\liminf A_n=\bigcup_{k=1}^\infty\left(\bigcap_{n\geq k}A_n\right)$ and $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$. ...
3
votes
2answers
57 views

Bounding a function on sets with a prescribed measure

In the proof of a lemma in a paper I'm reading the following is claimed. Let $f$ be a measurable function with $f(t) \geq 0$ for $0 \leq t \leq 1$ and $\int_0^1 f(t)\,dt = 1$. Let $k$ be a fixed ...
1
vote
1answer
437 views

Borel sigma field

Is the $ \sigma $ - field generated by $[a,b]$, $a,b \in \mathbb Q $ and $ \sigma $ - field generated by $(a,b)$, $a,b \in\mathbb Q^c $ identical? Are they also the same as Borel $\sigma $ - field. ...
1
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0answers
46 views

Original source on a result of convergence in measure and a.e.

A relatively well-known result in measure theory states that given a sequence $(f_n)_{n=1}^\infty$ of measurable functions from a $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ to $\mathbb{R}$ ...
4
votes
0answers
69 views

Haar measure on transversals

It is well known that there exists an invariant Haar measure on Locally compact group $G$. Haar measure on coset space and double coset space with respect to a closed subgroup $H$ and a compact ...
0
votes
1answer
34 views

Outer measure $u^*$ generated by $(\varepsilon, \rho)$

Let $X$ be infinite and $f:X\rightarrow[0,\infty]$ an arbitrary function. Let $\varepsilon=\{\emptyset,X\}\bigcup\{\{x\}:x\in X\}$ and set$\rho(\emptyset)=0$, $\rho(X)=+\infty$ and ...
2
votes
3answers
84 views

Trying to calculate the probability that one RV exceeds another RV

I am running into a silly mistake when trying to calculate the probability that a random variable, $U$ is less than another random variable, $V$. I am hoping that someone can help me spot my mistake. ...
3
votes
0answers
48 views

Representations of a C*-algebra of bounded Borel functions

Let $X$ be a compact Hausdorff space. Let $B(X)$ be the C*-algebra of bounded Borel measureable functions on $X$ (under the supremum norm). I am curious whether the (say unital) $*$-representations of ...
2
votes
1answer
249 views

question about Dynkin's π-λ Theorem

I have seen in class Dynkin's $\pi$-$\lambda$ Theorem (http://en.wikipedia.org/wiki/Dynkin_system ) One of the applications of it that we saw is the following (these are my notes) : We consider ...
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0answers
77 views

confusion about the definition Hausdorff measure

3 down vote favorite I am studying real analysis and I have some problems in understanding properties of Hausdorff measure. Let $\mathcal{E}_\delta$ be collection of subsets of $\mathbb{R}^N$ whose ...
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vote
1answer
46 views

Measure of elements appearing infinitely many times

Let $A_1,A_2,\ldots$ be Borel sets in the interval $[0,1]$, and let $\mu$ be a countably additive measure. Let $\epsilon>0$. Suppose $\mu(A_n)>\epsilon$ for all $n=1,2,\ldots$. Is it ...
4
votes
2answers
159 views

Show $(\Omega, \mathscr{F}, P)$ is a probability space.

Show $(\Omega, \mathscr{F}, P)$ where $\mathscr{F}=\{\text{all subsets of }\mathbb{R}\text{ such that either }A^c\text{ or }A\text{ is a countable set}\}$, and $P(A)=0$ if $A$ is countable, $P(A)=1$ ...
3
votes
2answers
144 views

Lebesgue measure of a set and its closure

Is it true that for any $A\subset\mathbb{R}$ then the Lebesgue measure $$m(A)=m(\overline{A})$$ And why? where $\overline{A}$ denotes the closure of the set.
3
votes
1answer
125 views

Extending integrals from continuous functions to bounded Borel functions

Let $X$ be a compact, Hausdorff space. Let $B(X)$ be the Banach space of bounded, Borel-measurable, complex-valued functions on $X$ under the uniform norm. Let $C(X) \subset B(X)$ be the closed ...
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votes
0answers
133 views

Prove of inverse of function is sigma algebra in measurable space

Let R be a set and (S; B) be a measurable space. Let X : R -> S be a function. Show that the collection of all subsets of R given in f={Inverse of X(A) : A belongs to B} is a sigma algebra. I can ...
2
votes
1answer
124 views

Expectation conditioned on a sub sigma field

Let $X$ and $Y$ be two integrable random variables defined on the same probability space $(\Omega,\mathcal F,\mathbf P)$ Let $\mathcal A$ be a sub-sigma-field such that X is $\mathcal A$-measureable. ...
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1answer
150 views

When do Inverse Images of Measurable Sets Generate the $\sigma$-algebra?

Just out of curiosity! Let $(X,\mathcal{M})$ and $(Y,\mathcal{N})$ be measurable spaces and let $f: X\to Y$ be a measurable function. If $\mathcal{N}$ is generated by some collection of sets ...
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vote
0answers
178 views

Lebesgue vs Riemann: Trouble understanding a theorem on improper integrals

In the book "Measures, Integrals and Martingales" by R. Schilling, the Riemann integral is compared to the Lebesgue integral (Chapter 11). I have trouble verifying the following statement for myself ...
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0answers
82 views

Measure on cylinder sets

Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$ and let $C_A=\{(x,y)\in\mathbb{R}^2;x\in A\}$. Such sets are called cylinder sets. (i) Show that the collection of these sets forms a ...