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

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72 views

Why outer measure is defined by $m_n^*(A)=\inf \bigg\{ \sum_{i=1}^\infty l(I_i):\mathbb{F}=\{I_1,I_2,… \}\mbox{ is Lebesgue cover of } A \bigg\} $?

My questions is: Why the outer measure $m_n^*(A)$ is defined by $$ m_n^*(A)=\inf \left\{ \sum_{i=1}^{\infty} l(I_i): \mathbb{F}= \{I_1,I_2,\cdots \} \mbox{ is Lebesgue cover of } A \right\} $$ for ...
-1
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1answer
58 views

How to show that $A \subset B$?

Let $A, B \subset \mathbb{R}$ and $A=\left\{x \in \mathbb{R}: x \in B\left(\frac{b+a}{2},\frac{b-a}{2}\right) \right\}$ and $B=\left\{x \in \mathbb{R}: x \in B\left(\frac{d+c}{2},\frac{d-c}{2}\right) ...
-1
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1answer
78 views

Find the outer measure of set of rationals in [-1,1]

What is the outer measure of the set of rationals in [-1,1]? I've read that it is infinity? I believe that it is countable. Then why it is infinity?
1
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1answer
124 views

Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure

Let $B_t:$ 1-dimensional Brownian motion, $P:$ its distribution on the Wiener space $C([0,1],\mathbb{R})$ $X_t=aB_t+bt\text{; }t \in [0,1]$, $P_{a,b}$ its distribution $Y_t=a(t)B_t\text{; }a:[0,1] ...
2
votes
1answer
258 views

Why is it a Borel set?

Claim: $\displaystyle \bigcup_{a,b \in \mathbb{Q}}[a,b)$ is a Borel set. Solution: I arrive at $[a,b)=\displaystyle \bigcap_{a,b \in \mathbb{Q}}(a-\frac{1}{n},b)$. This means that $[a,b)$ is a ...
1
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1answer
163 views

Jensen's inequality and a estimate in $L^p$

In problem 3 we have: If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and $$\int_E e^{|f(x)|}dx =1,$$ then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and ...
3
votes
2answers
910 views

Why $\sigma$-algebras represent information, and what information does $\sigma(X)$ represent?

I am confused about the notion of $\sigma$-algebras representing information and what information is contained in $\sigma(X)$ for a random variable $X$. Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is ...
4
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0answers
204 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
1
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0answers
262 views

Prove using the definition of Lebesgue outer measure

If $B = X \cup Y$, and $X$ and $Y$ are disjoint compact intervals, then $$m^*(B) = m^*(X) + m^*(Y)$$ (This is not always true for disjoint subsets of $\mathbb{R}$, but it is for intervals) I ...
11
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3answers
4k views

Is composition of measurable functions measurable?

We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
2
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0answers
184 views

Continuity in set functions

Let a function be defined as $f:(\Omega_1,\mathcal{F}_1)\rightarrow (\Omega_2,\mathcal{F}_2)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are $\sigma$-fields in $\Omega_1$ and $\Omega_2$ respectively. ...
6
votes
1answer
118 views

Given $(x_n)$, does there exist a measure such that $x_n=\int_0^1t^n d\mu$?

Let $(x_n)$ be a sequence of real numbers. Does there exist a measure $\mu$ on $[0,1]$ such that $x_n=\int\limits_0^1t^nd\mu$ ?
5
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1answer
208 views

$\delta$-fined division and Lebesgue measure $\mu$

Let $E$ be a measurable subset of $[0,1]$. Then we know that we can choose an open set $G$ and a closed set $F$ such that $F\subset E \subset G \subset [0,1]$. For each $t\in [0,1]$, define ...
3
votes
3answers
164 views

Property holds everywhere except on some subset of arbitrarily small measure is equivalent to holding a.e.?

Suppose a property $P$ may or may not hold on a measurable subset of a measure space with measure $\mu$. Is for every $ε > 0$, there exists a measurable subset $B$ such that $μ(B) < ε$, and ...
13
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2answers
606 views

Is the intersection of two countably generated $\sigma$-algebras countably generated?

Let $\mathcal{A}$ and $\mathcal{B}$ be two countably generated $\sigma$-algebras on the set $X$. Is the $\sigma$-algebra $\mathcal{A} \cap \mathcal{B}$ necessarily countably generated? I suspect that ...
3
votes
2answers
289 views

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
2
votes
2answers
63 views

Integration a function of a single variable over a 2-dim measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be two measurable spaces and let $\mu$ be a finite measure on the product $\sigma$-algebra $\mathfrak B(X)\otimes \mathfrak B(Y)$. Let $f:X\to\Bbb R$ ...
1
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1answer
147 views

Outer Measure on $\mathbb{R}$

Let $p\ge0$, $d>0$ and $X \subset \mathbb{R}$. (i) Define: $$H_{p,d}(X) := \inf\left\{\sum_{i=1}^{\infty}\operatorname{diam} \left(\frac{A_i}{2}\right)^p : X \subset \bigcup_{i=1}^{\infty}A_i, 0 ...
1
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0answers
222 views

Inner measure from premeasure on a ring?

In the proof of Carathéodory's extension theorem, an outer measure $\mu^*$ is constructed from a premeasure $\mu_0$ on a ring $A$ over an arbitrary set as $$\displaystyle ...
10
votes
2answers
2k views

“Scaled $L^p$ norm” and geometric mean

The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ ...
3
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0answers
237 views

Show a locally integrable function vanishes almost everywhere

Let $u\in L^{1}_{loc}(\Omega):=\{f:\Omega \to \mathbb R\;| \int_{K}|f(x)|dx<\infty,\;K\subset\Omega\; \mathtt{compact}\} $, where $\Omega\subset\mathbb{R}^{n}$, and let $\phi$ be a test function ...
1
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1answer
873 views

Prove that the indicator function for $\mathbb{Q}\cap[0,1]$ is not Riemann integrable

Prove that the function $ \phi_\mathbb Q : [0,1] \to \{0,1\}$ defined by $$\phi_\mathbb{Q}(w)=\begin{cases} 1 &\text{if } w\in \mathbb Q, \\ 0 & \text{if } w\notin \mathbb ...
5
votes
1answer
498 views

Integral of simple functions in standard and non-standard representation

Some definitions Let $(X,\mathbb X,\mu)$ be a measure space. A real-valued function is simple if it has only a finite number of values. A simple $\mathbb X$-measurable function $\varphi$ can be ...
1
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1answer
65 views

Constant $M$ bounding $|f|$ except on a set of 'small' measure

Let $f$ be a an extended real-valued measurable function on the interval $[a,b]$ that takes values at $\pm \infty$ on a set of measure $0$. Can we always have a constant $M$ such that $|f| \leq M$ ...
0
votes
0answers
54 views

Can random variables always be identified with their distributions?

This is a question coming from a discussion between Ilya and me (Thanks, Ilya!): a random variable standalone is exactly a representation of a distribution, nothing more. However, when it ...
0
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1answer
78 views

Countable additivity and Lesbesgue-Stieltjes measures

Let $\mathcal{F}_0(\mathbb{R})$ be the field of finite disjoint union of right semi-closed intervals. Let $F:\mathbb{R}\rightarrow \mathbb{R}$ be a non-decreasing right continuous function. Then the ...
2
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1answer
137 views

$f \in L^1 (\mathbf{R}) \iff\mathop{\lim_{a \to -\infty}}_{ b \to +\infty} \int_a^b |f(x)| \mathrm{d} x \text{ exists and it is finite}$

Let $(\mathbf{R}, M, m)$ be Lebesgue measure space and $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function. Show that $$f \in L^1_m (\mathbf{R}) \text{ if and only if } \mathop{\lim_{a ...
3
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1answer
727 views

How to prove this collection is a sigma algebra

This is one of the previous comp question. I would appreciate if somebody can give me a proof. Let $\mathbb A= \{E \subset X: E $ is a countable or $E^c$ is countable $\}$. To prove this collection ...
0
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1answer
174 views

show that a sequence of functions is bounded by an integrable function

show that the sequence-indexed with $a_n$ , $${1\over{1+t^2}} - {e^{-ta_n}\over{(1+t^2)}}(\cos a_n + t\sin a_n)$$ is bounded from above by an integrable function for a sufficiently large $a_n$ ...
0
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2answers
596 views

Approximating characteristic functions by continuous functions

The Urysohn Lemma is a very useful lemma,this lemma appears in several equivalent forms, one of them, what interests me is the following: Uyshon Lemma: For every closed set $K$ in $X$ and every open ...
6
votes
0answers
207 views

Product of complex measures

Let $\lambda, \mu$ be complex measures on $(X,\alpha)$ and $(Y,\beta).$ Prove there exist a unique complex measure $\lambda\times \mu$ on the sigma algebra $\alpha\otimes \beta,$ such that ...
0
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2answers
228 views

When is this function Lebesgue integrable (based on variables)?

I didn't know how to start with this basic question : Let $a,b > 0$ and $f$ a function : $$f(x)={xe^{-ax}\over 1-e^{-bx}}.$$ Find $a$ and $b$ so that $f$ is Lebesgue integrable in ...
4
votes
2answers
1k views

$f$ measurable with $f=g$ a.e. then $g$ measurable

How do I prove this proposition from Royden's Real Analysis: If $\mu$ is a complete measure and $f$ is a measurable function, then $f=g$ almost everywhere implies $g$ is measurable. In proving ...
3
votes
2answers
169 views

An inequality about sequences in a $\sigma$-algebra

Let $(X,\mathbb X,\mu)$ be a measure space and let $(E_n)$ be a sequence in $\mathbb X$. Show that $$\mu(\lim\inf E_n)\leq\lim\inf\mu(E_n).$$ I am quite sure I need to use the following lemma. ...
3
votes
1answer
102 views

Bound on the growth of a singular measure

Working through some measure theory, I came upon the Lebesgue-like decomposition for monotonic functions. In that context, I've cooked up a singular measure $\nu$ on $[0,1]$ about which I know only ...
2
votes
1answer
181 views

On the relation of a certain set function with the outer measure

Let $A$ be a set and define $m^{**}(A) \in [0,\infty] $ by: $m^{**} (A)= \inf\{m^{*}(\mathcal{O}) \ | \ A \subset \mathcal{O}, \mathcal{O} \ \text{open} \} $ How is $m^{**}$ related to the outer ...
2
votes
2answers
227 views

Existence of a measure-preserving mapping between two given measure spaces?

Given two measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i), i=1,2$, does there always exists a measure preserving mapping $(\Omega_1, \mathcal{F}_1, \mu_1) \to (\Omega_2, \mathcal{F}_2, \mu_2)$? One ...
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2answers
955 views

How to show that if $\inf A > - \infty$ and $\inf B > - \infty, A, B \subset \mathbb{R}$ then $A+B = \{a+b:a \in A, b \in B \}$ is bounded below?

How to show that if $\inf A > - \infty$ and $\inf B > - \infty$, $A,\, B \subset \mathbb{R}$ then $A+B = \left\{a+b:a \in , b \in B \right\}$ is bounded below?
2
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1answer
194 views

bounded measurable function

I'm looking at my old review questions from my real analysis notes from years ago, and see this problem: Let $g$ be an integrable function. on $[0,1]$. Is there a bounded measurable function $f$ ...
5
votes
2answers
203 views

extension of a non-finite measure

For a finite measure on a field $\mathcal{F_0}$ there always exists its extension to $\sigma(\mathcal{F_0})$. Can somebody give me an example of a non-finite measure on a field which cannot be ...
3
votes
2answers
125 views

Proof of Borel isomorphism theorem by Rao and Srivastava

Does someone know how Rao and Srivastava proved the Borel isomorphism theorem in their paper "An elementary proof of the Borel isomorphism theorem" published in 1995? I can't find anywhere this ...
4
votes
1answer
227 views

Price of Kolmogorov's extension theorem?

From Wikipedia an alternative way of stating Kolomogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure $\nu$ on ...
0
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1answer
120 views

Jordan-unmeasurable closed bounded subset of $R$

I know that such subset exists, and read somewhere that the Cantor set is an example of this (the Jordan measure is assumed). However, I couldn't find any proof to this, and really don't know if it's ...
2
votes
2answers
172 views

expectation value 3

Suppose that $X$ is a non-negative random variable and there exist constants $A,B$ such that $$\forall t > 0\colon P(X>\frac{1}{t})<Bt $$ and $$\forall t > 0\colon E(\sin(tX))<At$$ I ...
0
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1answer
103 views

Prove that set is measurable

I'm wondering how to solve this. We have space X, in which there is a sigma-field $M$. We have a sequence of measurable functions $f_n: X \rightarrow \mathbb{R}$. Let $A= \{ x \in X: $ sequence ...
3
votes
1answer
107 views

Coupling of two random variables

Let $X_1,X_2$ be two random variables on $\Bbb R$ with distributions $P_1$ and $P_2$, and let $\Bbb P$ be their coupling, $$ \Bbb P\circ X_1^{-1} = P_1,\quad \Bbb P\circ X_2^{-1} = P_2. $$ Is that ...
3
votes
1answer
82 views

Can the identity Lebesgue function be almost surely Borel?

I am trying to construct a counterexample to a homework question, and my intuition says my example is good, but I couldn't prove it. Here is the problem: Let $f : \left[0, 1\right] -> \left[0, ...
3
votes
3answers
3k views

Can someone explain the Borel-Cantelli Lemma?

I’m looking for an informal and intuitive explanation of the Borel-Cantelli Lemma. The symbolic version can be found here. What is confusing me is what ‘probability of the limit superior equals $ 0 ...
1
vote
4answers
474 views

Is there an abbreviation for “almost all $x\in X$”?

Is there an abbreviation for "almost all $x\in X$? I have "$\forall a.e. x\in X$" in my mind, but i see nobody uses this..
5
votes
1answer
107 views

Upcrossing measurable

For upcrossings we have defined $S_{n}=\inf \{k>B_{n}:x_{k}>b\}$ and $B_{n}=\inf \{k>S_{n-1}:x_{k}<a\}$. The number of upcrossings over the interval $(a,b)$ is given by ...