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

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2
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0answers
46 views

Existence of invariant Borel measures in infinite-dimensional separable Banach spaces.

Let $\mu$ be a Borel measure in an infinite-dimensional separable Banach space $X$. We say that $h \in B$ is admissible(in the sense of quasiinvariance) for the measure $\mu$ if measures $\mu$ ...
2
votes
1answer
63 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} -...
0
votes
1answer
44 views

There exists a measure such that the sum of derivatives is the integral

This is a homework question in functional analysis. If $n \geq 1$, show that there is a measure $\mu$ on $[0,1]$ such that $\displaystyle\sum_{k=1}^n p^{(k)} \left( \dfrac{k}{n} \right) = \int p \,...
8
votes
4answers
233 views

Does the operator $T(f)(t) := f(t) - f(0)$ preserve measurability?

Denote by $\mathcal{B}$ the Borel field on $\mathbb{R}$, denote by $\mathbf{C}_{\left[0,\infty\right)}$ the set of continuous, real-valued functions over the domain $\left[0,\infty\right)$ and denote ...
0
votes
1answer
47 views

convergence ( in measure theory)

Does $g_n=n \mathcal{X_{[1/n,2/n]}}$ converge to g=0. Leb. measure My idea is since the $[1/n,2/n]$ goes to zero as n goes to infinity, there is no element in the interval, so the charateristic ...
0
votes
2answers
69 views

Lebesgue measure identity

Let $A,B\subset \Bbb R$ non-empty. Let $E,F\subset \Bbb R$ measurable such that $A\subset E ,B\subset F$ and $m(E\cap F)=0$. Then $m^{*}(A\cup B)=m^{*}(A)+m^{*}(B)$. I need help with the proof of the ...
0
votes
1answer
88 views

Not countable generated sigma field

I need to show that $ F=(A \in \Omega $| A countable or co-countable) with $\Omega$=(0,1] is not countable generated. I have started supposing that F is countable generated and I have a hont that tell ...
0
votes
1answer
24 views

A basic doubt on partial derivative and integrability

I want an example of a function $f(x,y): \Bbb R^2 \to \Bbb R$ such that for each $y$, $f(x,y)$ is integrable w.r.t $x$, but the partial derivative w.r.t $y$ is not integrable w.r.t $x$.
0
votes
2answers
157 views

Borel measurable function

I'm struggling on the following question from a past paper: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined ...
1
vote
1answer
98 views

Lebesgue outer measure satisfies $\lambda^{*}([a,b]) \leq b-a$ [duplicate]

Aaagain, I fail to understand the trivial: Using compactness argument it is straightforward to show: $$\lambda^{*}([a,b]) \geq b-a$$ And everything is OK. But, regarding $\lambda^{*}([a,b]) \leq ...
9
votes
1answer
204 views

Significance and applications of the Riesz Representation Theorem in locally compact Hausdorff spaces

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
2
votes
1answer
87 views

How many algebras are there of finite-sized $\Omega$?

An algebra of $\Omega$ is a family that contains $\Omega,$ is closed under complement and finite union. If $\Omega=\{\}$, there can only be 1 algebra associated with $\Omega$: $2^{\Omega}.$ If $\...
1
vote
1answer
47 views

I don't understand why $\mu$ is not complete

Let $X$ be a set with at least two element. Consider the $\sigma$-algebra on $X$ as $\mathcal A=\{\phi,X\}$ and a measure on $\mathcal A$ defined by $$\mu(\phi)=0\\\mu(X)=1$$ My note ($2.12$) says $\...
0
votes
1answer
110 views

Need a proof for an assumption on conditional probability density function based on probability theory

While reading book "Elements of Information Theory", I came across an assumption used in a proof on page 33. The assumption is as follows. Let $(X,Y)\sim p(x,y)=p(x)p(y|x)$. "If $p(y|x)$ is fixed, ...
3
votes
1answer
130 views

Pushforward via $(x,y,z) \mapsto xy+(1-x)z$ of Lebesgue measure

Is there any easy way to see that the map $f(x,y,z) = xy+(1-x)z$ pushes forward the (three-dimensional) Lebesgue measure on the unit cube to the (one-dimensional) Lebesgue measure on the unit ...
4
votes
1answer
111 views

Set of independent sets is a sigma-algebra?

I was wondering whether the set $\{E \in \mathscr{E};A,B,E \text{ are independent} \}$ is always a sigma-algebra, when $A,B$ are two independent sets in some probability space? I know that it is a ...
0
votes
1answer
117 views

A question about Lebesgue measurable sets

Let $E\subset\mathbb{R}$ be a measurable set, and define $F=\{(x,y):x-y\in E\}$. Is $F$ a measurable set? why? Thanks to any one who give me a hint!
1
vote
0answers
46 views

question about sigma field

How do you determine the σ-field and the Dynkin system in Ω={1,2,3,4}? I know the smallest σ-field is {empty set, Ω}, and also I have worked out some others {empty set, {1},{2,3,4},{1,2,3,4}}, {...
0
votes
1answer
59 views

Why is this $\eta$-measurable?

So in a proof I was reading, it stated that if $\eta = P(\xi\in \cdot|\mathcal{F})$, then $\int f\, d\eta$ is $\eta$-measurable. Why is this true? Here, $\mathcal{F}$ is a sigma-algebra and $\xi$ is ...
3
votes
1answer
41 views

Without using $m^*(\cup_{i=1}^\infty A_i)=\lim_{n\to\infty}m^*{(A_1\cup…\cup A_n)}$ how to prove it?

Without using $$m^*(\bigcup_{i=1}^\infty A_i)=\lim_{n\to\infty}m^*{(A_1\cup...\cup A_n)}$$ how to prove it? $m^*$ is the Lebesgue outer measure. Linked: Is $\mu(\cup A_n)=\sum\mu(A_n)$ an ...
4
votes
2answers
168 views

Is $\mu(\cup A_n)=\sum\mu(A_n)$ an overstatement?

In defining a positive measure $\mu$ over an abstract measure space $(X,\mathcal A)$ isn't saying for any countable pairwise disjoint collection $\{A_n\}\subset\mathcal A,~\mu(\cup A_n)=\sum\mu(A_n)$...
2
votes
0answers
121 views

Expectation conditioned on an event and a sigma algebra

Suppose we have a probability space $(\Omega, \mathcal{F}, P)$ and $\mathcal{F}$-measurable random variables $X$ and $Z$ on this space. Suppose $Z$ is discrete and let $\mathcal{G}=\sigma(Z)$ The ...
5
votes
1answer
315 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
2
votes
0answers
28 views

Distributions with a given mean and covariance

Fix $X := \mathbb R^d$ for some $d \ge 1$. Fix a vector $m \in X$ and a covariance operator $k : X^* \to X$, i.e., a symmetric, nonnegative-definite operator. Let $\Delta_{m,k}(X)$ denote the ...
3
votes
2answers
118 views

Lebesgue integral question from wiki

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: Let $f: \...
2
votes
2answers
317 views

Books on measure theory

Can someone kindly suggested a good book on measure theory? Taking into consideration a good treatment of the abstract measures and Caratheodory approach.
1
vote
1answer
218 views

Does there exist a Lebesgue nonmeasurable set whose measurable subsets are all null sets?

Let $\mu$ be the Lebesgue measure. Let $A$ be a Lebesgue-nonmeasurable set. Define $S:=\{E\subset A : E\text{ is Lebesgue measurable}\}$. Does there exists a nonmeasurable set $A$ satisfying $\...
0
votes
1answer
20 views

Confusion with notation P[event in dy]

What does it mean when people write: $\Pr \{ B \in dy \}\ $ What I don't understand is what $ dy$ in there means.
2
votes
0answers
191 views

Measures with bounded total variation norm compact in $M(X)$?

Let $X$ be a separable, metric, compact space. (e.g. an interval in $\mathbb{R}$ like $[0,10]$). Let $M(X)$ be the set of all finite signed measures over $X$ with weak-*-topology (in probability ...
2
votes
0answers
63 views

How to count Dynkin system for finite sets?

For a set of finite elements, is there a good way to list all of its Dynkin systems, please? I understand that all $\sigma$-algebras of a set are also Dynkin systems. Therefore, we should as many ...
2
votes
1answer
88 views

sigma finite measure space

Let $(X,\mathcal{S},\mu)$ be a $\sigma$-finite measure space and $\phi:X\rightarrow X$ be a measurable transformation (in the sense that $\phi^{-1}(A)\in\mathcal{S}$ for each $A\in\mathcal{S}$). For ...
1
vote
1answer
169 views

A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...
0
votes
1answer
64 views

Showing that two functions defined as limits related to partitions are equal almost everywhere

Given a bounded function $f: [a, b] \rightarrow ℝ$, let $H(x)=\lim_{\delta \rightarrow 0} \sup_{|y-x| \leq \delta} f(y)$, $h(x)=\lim_{\delta \rightarrow 0} \inf_{|y-x| \leq \delta} f(y)$; $g = \...
0
votes
0answers
32 views

Bound on difference of measure

I am considering the following setting: Let $f,g$ be sufficient 'regular' function, and $A=\{\Delta f<0\}$ and $B=\{\Delta g<0\}$, what I want is that if $||f-g||<\delta$ in some norm, then $\...
0
votes
3answers
234 views

Properties of the Characteristic/Indicator Function

Let $B_1,B_2,...$ be a countable family of disjoint subsets of $\Bbb R^d$. For any set $E \in \Bbb R^d$, let $\chi_E (x)=1$ if $x \in E$ and $\chi_E (x)=0$ otherwise. Is it true that $\chi_{\bigcup ...
2
votes
1answer
122 views

For what reason, the surface measure represents the surface area?

Let $||\cdot||$ be a norm on $\mathbb{R}^n$ (It's an arbitrary norm, not 2-norm) Define $S^{n-1}=\{x\in\mathbb{R}^n : ||x||=1\}$ Let $\mu$ be the n-dimensional Lebesgue measure. Define $\sigma(E)=n\...
1
vote
1answer
75 views

Some property of conditional independence

Given random variables $Y, Z, X_1, X_2$ Is there some relation between $Y $ and $Z$ are conditionally independent given $(X_1, X_2)$ $Y $ and $Z$ are conditionally independent given $X_1$, and ...
1
vote
1answer
58 views

spectral measure of non-empty and open set is non-zero proof

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf Part d) on page 322 and his proof appears on page 324. I didn't quite understand his proof so I had a go at ...
2
votes
1answer
1k views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
3
votes
1answer
141 views

Constructively generating a sigma algebra

We have a collection $\mathcal{C}$ of sets (includes $\Omega)$ and would like to constructively generate the sigma algebra $\sigma(\mathcal{C})$. Would the following process work? Let $\mathcal{S}=\...
2
votes
2answers
149 views

Proving a particular function is a measure

Statement Let $(X,\Sigma)$ be a measurable space. Let a function of sets $\mu:\Sigma \to \mathbb R_{\geq 0}$ that satisfies: $A, B \in \Sigma \space \wedge \space A \cap B = \emptyset \space \...
0
votes
1answer
113 views

Necessary and sufficient conditions for countably additive probability measure

I have been struggling with understanding the solution to the exercise that follows. I cannot grasp why one of the conditions is that $x+y+z=2$. How do they come up with the value 2? Exercise: ...
2
votes
0answers
52 views

If $A$ is a nonmeasurable subset of $X$, and $E$ is measurable, s.t. $A\subseteq E$, then $\mu(E\setminus A)>0$

Let $\mu$ be an outer measure on a set $X$, If $A$ is a nonmeasurable subset of $X$, and $E$ is measurable, s.t. $A\subseteq E$, then $\mu(E\setminus A)>0$. Now i don't ask, how to prove it. We ...
0
votes
1answer
52 views

How do i prove that a dilation(?) of a Borel set is a Borel set?

Define $\mathbb{R}^+ \cdot A = \{rb\in\mathbb{R}^n:r\in (0,\infty) , b\in A\}$, $\forall A\subset \mathbb{R}^n\setminus\{0\}$ Let $\mathscr{B}$ denotes the Borel-algebra. Let $A\in\mathscr{B}_{\...
0
votes
1answer
52 views

invariant subspace partition

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf on page 327 Rudin says that M and M' are invariant subspaces. I'm guessing he means non-trivial so how does ...
2
votes
0answers
58 views

Bochner measurability

I have the following problem. Let $(\Sigma, \Omega, \mu)$ be a measure space and let $X$ be a Banach space. Take a function $f \colon \Omega \rightarrow \mathbf{B}(X)$, which takes values in space of ...
0
votes
1answer
85 views

Understanding the definition of the empirical measure

I'm reading Kosorok's Introduction to Empirical Processes and Semiparametric Inference and I'm stuck on an important definition. We define the empirical measure to be $P_n=n^{-1}\sum_{i=1}^n\...
3
votes
1answer
733 views

limit of increasing sequence of measures is a measure

Statement Let $(X,\Sigma,\mu)$ be a measurable space and let $(\mu_n)_{n\geq 1}$ be a sequence of measure in this space. Suppose that this is a monotone increasing sequence, in the sense that $\mu_n(...
4
votes
0answers
1k views

Meaning of “almost everywhere” in measure theory.

I'm slightly confused about the term almost everywhere as it is used in Folland's real analysis. Given a measure space $(X, \mathcal{M}, \mu)$ Suppose $f \equiv g$, $\mu$-almost everywhere where $f, ...
1
vote
1answer
86 views

Measurable vector bundles trivial

I hope you can help: If $E$ is a measurable vector bundle over a compact metric space $(X,\mu)$ then there is a subset $Y\subset X$ such that $\mu(Y)=1$ and $\pi ^{-1}(Y)$ is isomorphic to a trivial ...