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

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2
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69 views

Decomposing Countable Union of Measurable Sets

Why can every set $E$ in the real numbers with $\mu^{*}(E)=\infty$ be realized as the disjoint union of countably many measurable sets, each of which has finite outer measure? I'm trying to see this ...
4
votes
1answer
495 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
0
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2answers
129 views

Complex Measures: Decomposition

Given a complex measure: $\mu:\Sigma\to\mathbb{C}$. Consider its decomposition into positive measures: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ ...
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0answers
41 views

Neighborhood near $0$ for a sequence of non-negative measurable functions can be made arbitrarily small in measure.

I am doing some practice exercises from Avner's FoMA (2.4.1 to be precise). And encountered the following: Let $(X,\Sigma, \mu)$ be a finite measure space. Let $\{f_n\}$ be a sequence of a.e. ...
1
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1answer
67 views

Algebra Generated by Open and Closed Intervals

If $E$ is the collection of all open intervals $(a,b)$ in $X=[0,1]$, how do I know that the $\sigma(E)$ contains all closed intervals $[a,b] \subset X$, in particular closed intervals involving the ...
2
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1answer
33 views

signed measure and expected value

Let $(\Omega, F, \mathbb{P} )$ be a probability space. Let $\mathbb{G}$ be a sub-algebra of $F$ . If $X: \Omega \rightarrow \mathbb{R}$ is a random variable with $\mathbb{E}[|X|] < \infty$. If ...
0
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0answers
77 views

Notation of measures $d \mu$

I am reading the paper http://www.ams.org/mathscinet-getitem?mr=3246935 and there some notation that I have found a bit confusing on page 1503 between Lemma 4.2 and 4.3. I'll give as much context as I ...
3
votes
1answer
330 views

Union of uncountable measurable sets is bounded?

I'm trying to prove that if $ \{A_r\; :\: r>0\}$ is a family of measurable subsets of $\mathbb{R^n}$ such that $A_r \subset A_s$, if $r<s$, then uncountable union $\cup_{r>0} A_r$ is ...
1
vote
1answer
37 views

Sub-additivity in a probability space

I am proving th generalization of sub-additivity in probability. And i cant proceed to the next step because i cant show that this is true: $P( \bigcup A_i - \bigcup B_i )= P( \bigcup A_i) - P( ...
1
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1answer
48 views

Help understanding the proof of $\mu$-completion of a sigma algebra $\mathfrak M$

The theorem (from Rudin) states: For a measure space $(X,\mathfrak M, \mu)$, let $\mathfrak M^*$ be the collection of all $E\subset X$ for which there exists sets $A$ and $B\in\mathfrak M$ such that ...
3
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3answers
245 views

Important examples of measures which are not $\sigma$-finite

I think a measure which is not $\sigma$-finite is pain in the ass. I wish I could safely assume all the measures are $\sigma$-finite. I wonder if my wish is reasonable. Here's my question: what are ...
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1answer
40 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
2
votes
1answer
42 views

Measure on intersections of unions

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...∈\mathcal{A}$, assume that $\sum\limits_{j=1}^{\infty}=\mu (A_j)<\infty$ We have ...
2
votes
1answer
94 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
1
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0answers
49 views

The outer Lebesgue measure is actually an outer measure

I am looking at the proof that the outer Lebesgue measure that is defined by $$m^*(A)=inf\{\sum_{n=1}^{\infty}v(R_n):A \subset\cup_{n=1}^{\infty}R_n , ( \text{ where } R_n \text{ are open rectangles}) ...
2
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0answers
42 views

Covering in Sigma-Rings

Let $E$ be a collection of subsets in a space $X$. I would like to show that for all elements $A \in \sigma(E)$, the $\sigma$-ring generated by $E$, the exists a countable collection of subsets ...
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0answers
34 views

Determine integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$.

Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where ...
0
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1answer
60 views

the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space $(\Omega,\mathcal{F})$, every function $f:\Omega \rightarrow R, f\geq 0$ can be written as the uniform limit of an increasing limit of ...
0
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1answer
128 views

diffeomorphisms preserve zero measure

Suppose $\Omega\subset \mathbb R^N$ is an open set and $f:\Omega\rightarrow f(\Omega)$ is a $C^1$ diffeomorphism. Show that if $F \subset \Omega$ has zero measure then $f(F)$ has zero measure. I ...
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1answer
50 views

Sequence of Measurable Functions defined on R

I'm struggling with this problem from my most recent homework assignment in measure theory. Let {$f_n$} be a sequence of measurable functions defined on $\Bbb R.$ Show that the set $E =$ {$x \in ...
0
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1answer
106 views

Open squares are a generator of the Borel sigma-algebra

Let K be the set of open squares in $\mathbb{R}^2$ i.e. $K=\{(a,a+r) \times (b,b+r):a,b \in \mathbb{R}, r>0\} $ I have to show that $\sigma (K) = \mathcal{B}(\mathbb{R}^2)$. I know that i have ...
0
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1answer
71 views

Intersections of measurable sets

Let $(X, \mathcal{A},\mu)$ be a measure space, and let $A_1,A_2,...\in \mathcal{A}$. Assume that $\sum_{j=1}^\infty \mu(A_j)<\infty.$ I have already proved following statements: (1) $n \cdot ...
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1answer
89 views

Examples of measures that induce certain inclusions in the Lp spaces.

I apologize for the terribly worded title, but I didn't know how else to title this questions (which comes from Rudin's Real & Complex Analysis chapter 3 questions). The question says: For ...
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1answer
66 views

Prove that measures on intersections =0

I am stuck with a rather pesky problem. I have to prove that for the measure space $(X,\mathcal{A}, \mu)$ $A_1,A_2.....\in \mathcal{A}$, $$\mu \left(\bigcap_{j=1}^{\infty}A_j\right)=0$$ and ...
0
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1answer
117 views

Complex Measures: Variation

I highly doubt a proof of mine for complex measures... First, a complex measure can be decomposed into positive measures: ...
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1answer
91 views

Complex Measures: Integration

Reference A rigorous treatment can be found in: Complex Measures: Integrability Problem Integration w.r.t. complex measure usually is defined via the Radon-Nikodym derivative: $$\int ...
0
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1answer
83 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
4
votes
1answer
269 views

Determining Measurability given property of symmetric difference

I came across this statement and I am not sure why it is true. If $\mu$ is sub-additive and $\mu(E\bigtriangleup F)$ is in the null set (where $E \bigtriangleup F$ denotes the symmetric difference ...
2
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0answers
39 views

Outer measure and Subadditivity

Consider $\{(a,b] \cap \mathbb{Q}:a\leq b, a,b \in \mathbb{Q}\}$, with $\tau((a,b] \cap \mathbb{Q})=b-a$, and $\mu$ the outer measure of $\mathbb{Q}$ generated by $\tau$. I want to determine $\mu(R)$, ...
3
votes
1answer
71 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
1
vote
1answer
91 views

measure theory, $\delta$-$\epsilon$

I am struggling with this exercise: Suppose that $f \in L^1(\Omega, A, \mu)$. Show that for each $\epsilon > 0$, there is a $\delta >0$ such that $\mu(E) < \delta \rightarrow\int_E|f| ...
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2answers
91 views

If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A

If $\mathcal{E}$ is collections of subsets of a set $X$ and let $A \subset X$ be a subset. Show that the generated $\sigma$-algebra of $\mathcal{(E \cap}$ A) =the generated $\sigma$-algebra ...
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1answer
45 views

Regarding measurability of the set of points of convergence

I have a question about an answer to this, it is to show $$\{ x \in X: \lim_{n\rightarrow \infty}f_{n}(x)\text{ exists and is finite}\}$$ is measurable, given $f_n$ is measurable and $X$ is a measure ...
0
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1answer
62 views

Non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability

I think non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability are pain in the ass. The major trouble is that the Haar measures on them are not necessarily ...
0
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2answers
43 views

Examples of subsets of measure zero

Let $(X,\mathfrak M,\mu)$ be a measure space (Here $X$ is the set, $\mathfrak M$ is the $\sigma$-algebra, and $\mu$ is the measure). Let $N\in\mathfrak M$ with $\mu(N)=0$. It may happen that $N$ may ...
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2answers
92 views

Approximating Measurable Functions

There are questions with such title on stackexchange, but I am posting this question, since I didn't find solution to my question from other similar questions. The following is a basic theorem in ...
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2answers
42 views

Why does the following set always have finite measure?

If $E$ is a measurable subset of $\mathbb{R}$, and the Lebesgue outer measure $m^*(E) = \infty$, then why does $$E_n = E \cap [-n,n], \ \ \ n \in \mathbb{N}$$ have finite measure? What if $n = ...
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1answer
38 views

Why is the following statement true?

I am studying the proof of a theorem in Real Analysis by Royden & Fitzpatrick. I don't understand one of the statements. Let $E$ be any set of real numbers. Assume $E$ is measurable. Why, if $E$ ...
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1answer
61 views

Uniform Limit of Nets Measurable?

Clearly, the pointwise limit of a sequences of measurable functions is measurable: $$f_n\text{ measurable}\implies f\text{ measurable}\quad(f_n\to f\text{ pointwise})$$ (Especially, this holds true ...
0
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1answer
133 views

Measurability of the floor function

Let $u(x)=⌊x⌋$, i.e the largest integer not greater than $x$ . Determine $\{u≥a\}$ for all $a\in \mathbb{R}$. Show that $u$ is Borel-measurable. Can anyone help me with this problem?
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1answer
118 views

How is this $\mu_0$ not a premeasure?

I have a homework problem that I've been stuck on for some time. Define $\mathcal{A}$ to be an algebra on $\mathbb{Q}$ generated by the set $S = \{ (a, b] \cap \mathbb{Q} : a, b \in \mathbb{Q} \}$, ...
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3answers
33 views

Which is the intersection?

I got stuck at the following exercise. Could you give m a hint? We have the increasing sets $A_n=\{n,n+1,n+2, \dots \}$. To what is the intersection of the sets $$\cap_{n=1}^{\infty} A_n$$ equal ?? ...
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2answers
59 views

How to prove that a function is left continuous

I cannot work out this problem even though it seems not that difficult. Could anyone kindly give me any hint? Thanks! If $f(x)$ is measurable on $E \subset \mathbb R$, then $$ \varphi (t)=m\big(\{x ...
3
votes
1answer
36 views

Give example where an outer measure is strictly less than the set function from which it is defined.

Let $K $ be a class of subsets of $X $ where for every subset $A $ of $X $ there is a sequence $\{E _n \} $ of sets in $K $ such that $A \subset \bigcup _{n=1 }^{\infty } E _n $. Let $\lambda$ be a ...
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vote
1answer
142 views

Lebesgue measure of an uncountable point-set

For a countable set, it is easy to construct a countable cover of arbitrary small intervals to show that the set has measure zero. Can this reasoning be extended to sets with uncountably many isolated ...
2
votes
2answers
59 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
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1answer
112 views

Total variation inequality liminf

I'm trying to understand the proof, given by t.b. here Space of Complex Measures is Banach (proof?) . In the course of the proof, the author uses the equality $\left|\left| \mu - \mu_{m} ...
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1answer
19 views

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable?

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable? I understand that the union of measurable of sets is also measurable. But I am wondering if there is ...
0
votes
1answer
30 views

I do not understand why $E_n$ is measurable in this proof

Suppose we have a sequence of measurable functions $\{f_n\}$ on $X$. Also, suppose that (a) $0\leq f_1(x)\leq f_2(x)\leq ...\leq\infty$ for every $x\in X$ (b) $f_n(x)\rightarrow f(x)$ as ...
1
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2answers
42 views

Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...