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

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algebraic sum of a graph of continuous function and itself - measure > 0 imply nonempty interior?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Then $G+G$ is compact: algebraic sum of a graph of continuous function and itself Borel or ...
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35 views

Is there a function whose derivative is not Lebesgue integrable on R? [closed]

Is there a function else whose derivative is not Lebesgue integrable on R? And why is it not integrable?
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28 views

Relation between uniform distribution and dense curves

I have a question on the relation between space-filling curves and the joint distribution of two independent uniform random variables. Consider the probability space $(\Omega, \mathcal{F}, ...
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1answer
30 views

Why the probability distribution of a uniform random variable is the Lebesgue measure?

Consider the random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ distributed as a uniform on $[0,1]$. The probability distribution function of $X$ is defined as a map $$ ...
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35 views

Does $m^*(E)=m(E\cap A)+m^*(E\cap A^c)$ always true?

Caratheodory says that $E\subset \mathbb R^d$ is measurable if $$m^*(A)=m^*(E\cap A)+m^*(E^c\cap A)$$ for all $A\subset \mathbb R^d$. To me a more natural way would have been to define it as : $E$ is ...
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0answers
55 views

Comparison test and DCT

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
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0answers
38 views

Basic measure theoretic definitions of random variables/probability distributions: codomain versus range?

I have the following question: in the collection of all measure theoretic definitions of random variables/probability distributions/cdf/pdf it seems to me that what is considered is always the ...
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1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
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1answer
34 views

Real Analysis, Folland problem 1.4.20 Outer measures

Exercise 20 - Let $\mu^*$ be an outer measure on $X$, $M^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, $\overline{\mu} = \mu^*|M^*$, and $\mu^+$ the outer measure induced by $\overline{\mu}$ ...
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2answers
77 views

$f,g$ coincide on $\mathbb N$ iff $f(x)=g(x)$ a.e.

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
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48 views

Measure theoretic definitions of random variables/probability distributions

I'm asking your help to answer questions (a),(b),(c) outlined in the summary below. The questions are so connected that I found difficult to ask them separately. Could you also let me know if you find ...
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1answer
108 views

Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal ...
5
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1answer
63 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
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1answer
26 views

Equality of Measures and Intuition

Caveat: I have no formal training in measure theory and am learning as I go. The concept in this question is puzzling me: Equivalent measures if integral of $C_b$ functions is equal I'll re-state ...
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0answers
37 views

Is there a name for this measure?

For any given set $X$, define a measure $u$ on $\wp(X)$ where for all $A \in\wp( X)$: $$u(A)=0\text{ if }A\text{ is countable, and }u(A)=\infty\text{ otherwise}$$
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1answer
45 views

Why is A unbounded? [closed]

Let $$ A=\bigcup\limits_{n=1}^{\infty}G_{n} $$ where $$G_{n}=\left(\frac{1}{n+1},\frac{1}{n}\right)$$ $$\mu(G_{n})=\frac{1}{n}-\frac{1}{n+1}$$ or ...
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0answers
21 views

What is a good way to think of Caratheodory Hahn theorem intuitively?

I am reading the theorem in Royden's book on Caratheodory Hahn theorem. After reading it for like 10 times, I still do not quite understand it. Can anyone please offer some intuition?
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1answer
59 views

Sum of measurable functions is measurable: countable choice required?

The standard proof that the sum of measurable functions is measurable uses countable choice, via the countable subadditivity of outer measure ($\implies$ measurable sets are closed under countable ...
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1answer
36 views

Definition of outer Measure

As I understand it, the outer measure $\mu^{*}(A)$ is used to find the length of the smallest cover that covers $A$. However, in another definition, the outer measure is defined as the largest lower ...
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0answers
31 views

Lebesgue-Stieltjes integral and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
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1answer
64 views

Limit of a sequence of Lebesgue integrals

Let $ f\in L^{1}(E) $ and $ \{E_n\}$ be a sequence of measurable subsets of $E$. If $$ \lim_{n\to +\infty} m(E_n) = 0$$ prove that $$ \lim_{n\to +\infty} \int_{E_n} f = 0.$$ I tried to interchange ...
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0answers
32 views

Is the composition of uniformly distributed functions uniformly distributed?

Let $\mathcal{I}:=[0,1]$. Def: A measurable function $\varphi:\mathcal{I}\rightarrow \mathcal{I}$ is said to be uniformly distributed with respect to the Lebesgue measure $\Lambda$ if, for any ...
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1answer
27 views

The composition of measurable function is not measurable: only for Lebesgue-measurability?

Let $\mathcal{I}:=[0,1]$. Let $\mathcal{R}(f)$ denote the range of a function $f$. Let $\Sigma$ be the $\sigma$-algebra of $\mathcal{I}$. Consider the measurable and continuous functions ...
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1answer
40 views

Probability measures equipped with weak topology, is $P\mapsto P(A)$ measurable?

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the set of probability measures on $X$ equipped with the topology of weak convergence. So $\mathcal{M}(X)$ itself can be viewed as a compact ...
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1answer
47 views

Real Analysis, Folland problem 1.4.19 Outer Measures

Background information: Exercise 18 - Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and $\mathcal{A}_{\sigma\delta}$ ...
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1answer
43 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
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1answer
25 views

Uncountable null sets in arbitrary measure spaces

Does every measure space $(X,\Sigma,\mu)$ with $X$ and $\Sigma$ uncountable, have an uncountable null set?
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1answer
35 views

Using the monotone convergence theorem to show a function is integrable

Apply the monotone convergence theorem and the fundamental theorem of calculus to show that $f(x) = \left\{ \begin{array}{ll} x^{-a} & \mbox{if } 0 < x \leq 1 \\ \infty & \mbox{if } ...
2
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0answers
35 views

Prerequisites to understanding proof of Fubini's Theorem? [closed]

I'm currently studying tensor analysis, and I have studied elementary calculus (meaning calc I, II, III, and diffy Q), as well as linear algebra. Given all of this, what are the rest of the required ...
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55 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
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1answer
30 views

Does this set of infinite binary sequences have positive probability?

The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the ...
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1answer
49 views

Why is a set function equivalent to its induced outer measure if it is countably monotone?

Denote Outer measure as $\mu^*(E)$ and a set function $\mu(E)$. Define outer measure as the following: μ*(E) = inf $\Sigma\mu(E_k)$ where $\{E_k\}$ cover E. Why is a set function equivalent to its ...
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1answer
28 views

Can anyone explain the connections among ring, semiring, algebra, sigmaalgebra in the scope of measure theory?

Can anyone explain the connections among ring, semiring, algebra, sigma-algebra in the scope of measure theory and why are these concepts important?
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1answer
23 views

algebraic sum of a graph of continuous function and itself Borel or measurable?

Let $f\colon[0,1]\to\mathbb{R}$ be a continuous function. Let $G\subset\mathbb{R}^2$ be a graph of $f$. Does $G+G$ have to be: a Borel set? Lebesgue measurable?
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0answers
20 views

Finding an explicit Radon-Nikodym derivative

Let $(X, \mathcal{M}, \mu)$ be a measure space and let $f_{1}, f_{2} \in L^{1}(X)$. Define two signed measures $\nu_{1}, \nu_{2}$ on $(X, \mathcal{M})$ by $\nu_{1}(E) = \displaystyle\int_{E} f_{1} ...
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1answer
23 views

regularity of a measure

Let $\mathcal{A}$ be a $\sigma$-algebra containing the Borel algebra (everything is in a topological space). Let $m\colon\mathcal{A}\to[0,\infty]$ be a measure. The standard definition of regularity ...
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2answers
32 views

An example of a semiring which is not a ring.

In Bogachev's book on measure theory he states the following: The family of all intervals in the interval $[a,b]$ gives an example of a semiring that is not a ring. I am sure this is an easy ...
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1answer
21 views

Determining subsets in sigma field

I am given a sample space $= \{a, b, c, d, e\}$ and told that $\{\{a,b\}, \{b, d, e\}\}$ is a subset of the sigma field, which other subsets must the sigma field contain? I know the empty set and ...
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0answers
23 views

In which cases the outer Lebesgue measure is additive?

In which cases the outer Lebesgue measure is additive? it is known that $m^*(A\cup B)=m^*(A)+m^*(B)$ holds for disjoint bounded closed sets. 1) is it true for any disjoint closed sets? 2) is it ...
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1answer
57 views

1.9 from Rudin RCA

Hello! This is from Rudin's RCA. During my reading 1.9 I have couple question: $1)$ Why $(a)$ follows from Theorem 1.8 with $\Phi(z)=z$? If we put $\Phi(z)=z$ then $\Phi(u(x),v(x))=(u(x),v(x))$ but ...
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1answer
42 views

Integral of $f'$ where $f$ is continuous on $[a,b]$ and differentiable over $(a,b)$.

There is a problem which states that if $f$ is a function continuous on $[a,b]$ and differentiable almost everywhere on $(a,b)$ whose $|\text{Diff}_\frac{1}{n} f| \leq g$ almost everywhere on $[a,b]$ ...
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1answer
43 views

Real Analysis, Problem 1.4.17 Outer Measures [duplicate]

If $\mu^*$ is an outer measure on $X$ and $\{A_j\}_{1}^{\infty}$ is a sequence of disjoint $\mu^*$-measurable sets, then $\mu^*\left(E\cap\left(\bigcup_{1}^{\infty}A_j\right)\right) = ...
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1answer
28 views

$\overline{\int}_{E\cup F}f=\overline{\int}_{E}f+\overline{\int}_{F}f$ using ${\overline{\int}_{\Bbb{R}^n}}f=\inf\{\int h|\text{integrable }h\ge f\}$

Given $E$ and $F$ are disjoint, I have to show that $\overline{\int}_{E\cup F}f=\overline{\int}_{E}f+\overline{\int}_{F}f$ using ${\overline{\int}_{\Bbb{R}^n}}{f}=\inf\{\int h|\text{integrable }h\ge ...
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2answers
74 views

intersection of boundaries have Lebesgue measure 0

Suppose $A$ and $B$ are two sets in $R^n$ such that $\overline{A}\cap B \cup \overline{B}\cap A$ is empty then $\partial A \cap \partial B$ has $n$-dimensional Lebesgue measure $0$ (where $\partial ...
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0answers
14 views

Let G be a group with identity ε. If a, b ∈ Z and x ∈ G are such that x a = ε and x b = ε then show that x gcd(a,b) = ε. [duplicate]

I know the definition of the group is associative, inverse and identity. But, I have no idea where to start and how to solve it!
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0answers
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Jordan measure: half open intervals versus closed intervals.

I'm reading the definition of Jordan measure here and I have problems in understanding why the main definition should be given considering $n$-dimensional rectangles given by half-open intervals, ...
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1answer
13 views

Product of Lebesgue measure on Hilbert cube doesn't satisfy doubling condition?

The Hilbert cube $H$, is the infinite dimensional product $[0,1]\times [0,\frac12]\times...$ Let $\mu$ be product of Lebesgue measures $\mathcal{L}^1 \times \mathcal{L}^1\times...$, I heard that the ...
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1answer
20 views

Non-regular measure can be represented by a regular measure

Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular). Then the linear map $L : C_c(X) \to \Bbb C$ ...
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1answer
72 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
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0answers
19 views

If a set has finite outer measure, why can you find a countable number of sets that cover it?

Let $μ: S \to [0, \infty]$ be a set function defined on a collection $S$ of subsets of a set $X$ and $\bar{\mu}: M \to [0, \infty]$ the Caratheodory measure induced by $μ$. Let $E$ be a subset of ...