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

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36 views

Open Nullset containing $\mathbb{Q}$

I try to solve the following task: Let $\lambda_n$ be the Lebesgue measure in $\mathbb{R}^n$ and $\mathbb{Q}$ be the set of all rational numbers in $\mathbb{R}$. For any $\varepsilon > 0$ ...
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1answer
93 views

Measure Theory ; why it works? [closed]

Helllo I'm Studying measure theory ( Lebesgue and Fatou .. ) in University but I don't understand the utility of it ? i don't find any motivation to study it ; it seems complicated and i don't find ...
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2answers
104 views

Every $\sigma$ finite measure is absolutely continuous with respect to a finite measure.

Let $\mu$ be a $\sigma$- finite measure on $(X,M)$. Prove that there exists a finite measure $\lambda$ on $M$ such that $\lambda\ll\mu$ and $\mu\ll\lambda$. Can anyone give me a hint on how to start ...
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2answers
39 views

A nice property of L1 function in [0,1] .

Let f is in $L^1$$([0,1],m)$,$m$ is a Lebesgue measure and suppose that $f(x)>0$ for all $x$. Show that for any $ 0<\epsilon<1 $ there exists $\delta$ >0 so that $\int_{E} f(x)dx\ge \delta$ ...
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1answer
68 views

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}$ such that $m(E\cap(E+t)) = 0$ for all $t \neq 0$. Then prove that $m(E)=0$.

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}$ such that $m(E\cap(E+t)) = 0$ for all $t \neq 0$. Then prove that $m(E)=0$. I think that the function $f(t) = m(E\cap(E+t))$ is continuous but ...
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0answers
13 views

Composition of a linear continuous with Bochner integrable function is Bochner integrable

In Wiki, under section 'Properties', there is this sentence: If $T$ is a continuous linear operator, and $f$ is Bochner-integrable, then $T \circ f$ is Bochner-integrable and integration and $T$ ...
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2answers
56 views

What is the minimum number of points a sample space must contain that there exist n independent events that wouldnt have probability of zero or one

This is classic problem from most of the probability theory textbooks. What is the minimum number of points a sample space must contain in order that there exist $n$ independent events ...
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1answer
38 views

Moment generating function expansion proof

I am currently learning measure theoretic probability and I am trying to fill in the details for a sketch proof of the following well-known result: Let $X$ be a real-valued random variable on the ...
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1answer
30 views

Given a set of conditions, prove that there exists a measurable function

This problem comes from the book Real Analysis, by Folland section 2.2 problem 7, Background information: The problem says to use exercise 4 which is here and I will also provide the proof for If ...
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1answer
31 views

Proof: If event $A_1,\ldots,A_n$ are independent then their indicators are independent

Comes from Alan F. Karr Probability textbook page 81 $$ \prod\limits_{i\in I} P(A_i) \times \prod\limits_{j\notin J} [1-P(A_j]-\prod\limits_{i\in J} (P(A_j) \times \prod\limits_{j\notin J}[1-P(A_j)] ...
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1answer
30 views

Continuity of $\|f\|_p^p$ on the set where it is finite

Fix a function $f$ that is complex measurable on $X$, with the positive measure $\mu$ on $X$. Define $\phi:(0,\infty) \to [0,\infty]$ as $\displaystyle \phi(p) = \int_X |f|^p d \mu$, and define ...
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2answers
68 views

Probability that a random number belongs to $[a,b]\subseteq[0,1]$

Let $(a_k)$ be an infinite sequence of $0$s and $1$s chosen at random so that the probability that any $a_k=0$ is $1/2$. Show that the probability that the random number $\sum_{k=1}^\infty a_k/2^k$ ...
3
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0answers
42 views

Inequality involving homogenous function of degree -1

Let $p\in[1,\infty]$. For $f\in L^p(0,\infty)$ we define $Tf:x\mapsto \int_0^\infty K(x,y)f(y)\,dy$ where $K$ is homogenous of degree $-1$, i.e. $K(\lambda x,\lambda y) = \lambda^{-1} K(x,y)$ for ...
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1answer
123 views

Prove that $[0,1]^k$ in $\mathbb{R}^d$ has strict Hausdorff dimension $k$

I am trying to prove that the set $$[0,1]^k=\{(x_1,...,x_k,0,...,0):x_i\in[0,1]\}\subset\mathbb{R}^d$$ has strict Hausdorff dimension $k$. I have a hint: prove that there is a constant ...
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0answers
35 views

If $E\subset \mathbb R^n$ measurable then $aE$ is measurable for all $a\in\mathbb R$ and $m(aE)=a^nm(E)$

I want to show that if $E\subset \mathbb R^n$ is measurable then $aE$ is measurable for all $a\in\mathbb R$ and $m(aE)=a^nm(E)$. I don't really know how to do. Notice that $m$ is Lebesgue measure. My ...
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1answer
51 views

Properties of Hausdorff $s$-dimensional outer measure

I have some difficulties on proving results using the definitions of measures. This one is on Hausdorff outer measure. I need to prove the following intuitive result: Consider $s\geq 1$ and ...
2
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1answer
19 views

If $(\Omega, F, m)$ is a $\sigma$-finite measure space, show the following

The problem is Let $\mu$ be outer measure (extended) from $m$. Let $G$ be all $\mu$-measurable subsets of $\Omega$ (constructed using Caratheodory's Extension), $E\in G$, then show that there are ...
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1answer
63 views

Probability random binary number lies in an interval

Suppose we have a sequence $\{d_n\}$ where all the $d_n$ are either 1 or 0, with equal probability. Let $x=\sum_{n=1}^\infty d_n2^{-n}$. I need to show that $\mathbb{P}(x \in [a,b])=b-a$. I started ...
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0answers
45 views

If $X$ and $Y$ are independent random variables and $X$ is absolutely continuous then $X+Y$ is absolutely continuous

I'm stuck in the following problem. Let $X$ and $Y$ independent random variable where $X$ is absolutely continuous. Then $X+Y$ is absolutely continuous. This is what I have so far. Let ...
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0answers
21 views

Lebesgue integrable function composed with Continuously Differentiable Bijection

Assume $f:\mathbb{R}\to\mathbb{R}$ is a Lebesgue integrable function and let $\phi:\mathbb{R}\to\mathbb{R}$ be a continuously differentiable bijection. Show that: $$\int_\mathbb{R} f\circ ...
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2answers
63 views

Showing that $\frac{d\mu}{d\nu} = 1/\frac{d\nu}{d\mu}$

I am trying to prove the following statement: Suppose $\mu$ and $\nu$ are finite measures on the measurable space $(X,\mathcal A)$ which have the same null sets. Show that there exists a ...
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1answer
39 views

Hausdorff measure of a subset of $R^n$

I need to show if $d>n$ then $H^d(A)=0, A \subset R^n $ and is Borel. I have seen how it is done for an interval on $R^2$ but I am having troubles with he general case. Any help would be ...
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1answer
35 views

Does pointwise convergence imply that a subsequence converges in measure?

I am working to understand the relationships between the many modes of convergence. One of the true/false problems I am looking at is If $f_n\rightarrow f$ pointwise, then a subsequence ...
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1answer
22 views

Characterization Theorem of the Lebesgue Stieltjes Measure on the Real Line (Folland Theorem 1.18)

This is a theorem from Folland's Real Analysis, and I have difficulty understanding the proof. My problem is the last two sentences. $H_n$ as defined is clearly compact, and I follow the inequality ...
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2answers
22 views

probability densitiy function of $-X$

given a random variable $X$ with density $f_X(x)$. what is the density of $-X$? it's $$f_{-X}(x)=f_X(-x),$$ isn't it? is that obvious? is there a descriptive explanation?
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1answer
60 views

Questions on theorem 2.14 Rudin, real-complex analysis.

Some probably silly questions about such theorem, the proof is quite long and i don't understand some of the steps of this theorem. (i will probably update this post because of the length...). The ...
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1answer
86 views

If $a_{mn}\ge 0,\,$ then $\,\sum_{m\in\mathbb N} \sum_{n\in\mathbb N} a_{mn} = \sum_{n\in\mathbb N} \sum_{m\in\mathbb N} a_{mn}$. [closed]

I am studying Measure Theory, and in particular, integration of non-negative measurable functions. I have encountered the following problem: If $\,a_{mn}\ge 0,\,$ for all $\,m,n\in\mathbb N,\,$ then ...
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1answer
28 views

How to show that the closed unit ball is a measurable set for outer measure $m^*$?

How to show that $\{x \in \mathbb R^{n} : \|x\| \le 1 \}$ is an $m^* $-measurable set? I know that a set $E$ is said to be $m^*$-measurable (or Lebesgue measurable) if for any set $A \subset \Bbb ...
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1answer
32 views

Meaning of abs. value of a sigma field?

In a hand out I saw the a notation which looks to be the absolute value of a sigma field F, |F|. I googled it but I could not really find what it means and the notation confuses me. Anyone that might ...
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1answer
24 views

Use the Holder inequality to show that $f \ast g \in C_c(G)$

Let $G$ be a locally compact abelian group, and let $f \in L^p(G), g \in L^q(G)$. I'm trying to prove that $f \ast g \in C_0(G)$. The book I'm reading (Rudin, Analysis on Groups) gives the following ...
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1answer
13 views

Inequality on an integrable function

Suppose $f:X \to \mathbb{R}$ is an integrable function and $(X,F,m)$ a measure space. For all $C\in F$ with $m(C)>0$ $$\left| \frac{1}{m(C)}\int_C f dm\right|\le L$$ where $L>0$. Need to show ...
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1answer
62 views

Real Analysis, Folland problem 2.1.3 [closed]

If $\{f_n \}$ is a sequence of measurable functions on $X$, then $\{x:\lim f_n(x) \text{ exists}\}$ is a measurable set. Corollary 2.9: If $\{f_j\}$ is a sequence of complex-valued measurable ...
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0answers
45 views

$X,Y$ independent from sigma algebra $\Sigma$ then also $X+Y$?

Let $X,Y$ be independent from a sigma algebra $\Sigma$. Does this mean that $X+Y$ is independent from $\Sigma$? I just don't know how to show it, but maybe a yes/no answer and a good hint could help ...
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0answers
99 views

Chain rule for Radon-Nikodym derivative (without a.e.)

Assume that $\mu, \nu, \rho$ are $\sigma$-finite positive measures. If $\nu << \mu, \rho << \nu$, then $\rho << \mu$ and $$\frac{d\rho}{d\mu} = \frac{d\rho}{d\nu}\frac{d\nu}{d\mu}.$$ ...
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1answer
50 views

Measurability of a function taking values in the extended real line

Let $f:X\rightarrow \overline{\mathbb{R}}$ and $Y = f^{-1}(\mathbb{R})$. Then $f$ is measurable if and only if $f^{-1}(\{-\infty \})\in M$, $f^{-1}(\{\infty\})\in M$, and $f$ is measurable on $Y$ ...
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2answers
45 views

Construction of Outer Measures

I am trying to understand the following result. I know that there are a lot of questions about this here, but none addresses the point I am stuck on. Let $\mathcal E\subset\mathcal P(X)$ and take ...
2
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1answer
52 views

Independence of Random Variables By Guessing

Our lecturer said that if two random variables are independent, it should usually be "obvious" from their joined density. On the following examples, he then indeed proceeded to prove independence by ...
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0answers
36 views

Measure of the intersection of open sets

I try to solve the following task: Let $\lambda_n$ be the Lebesgue measure in $\mathbb{R}^n$. Consider an arbitrary subset $A \in M_{\lambda_n^*}$ such that $\lambda_n(A)<\infty$. Show that there ...
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1answer
102 views

Is every compact subset of $\Bbb{R}$ the support of some Borel measure?

I have tried to prove the exercise 2.12 in Rudin's RCA: 12 Show that every compact subset of $\Bbb{R}$ is the support of a Borel measure. For perfect (i.e. no isolated point) compact $K$ with ...
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1answer
48 views

How to prove that $m\left(\lbrace f>\alpha\rbrace\right)$ is a right-continuous function of $\alpha$?

Let $f$ be a Lebesgue measurable function that is finite almost everywhere on $\left[a,b\right]$. Prove that $m\left(\lbrace f>\alpha\rbrace\right)$ is a right-continuous function of $\alpha$, and ...
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1answer
95 views

How to calculate Lebesgue Outer Measure on $\mathbb{R}^2$

I am having some trouble on the steps I need to follow to calculate Lebesgue measure of some simple sets. For example, I am working on these two examples: $$1. ...
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1answer
24 views

Support of a Set in sigma algebra

Let $I$ be a large index set (you can think I being the set of reals), and for each $i \in I$ we have a $\sigma$-algebra $\Omega_i$ on the set $X_i$. Consider the Cartesian product $X =\prod_{i\in ...
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3answers
36 views

Does a point-wise convergence of measurable functions, imply a convergence in finite measure?

This is not true for infinite measures (Pointwise convergence, but not in measure). Is it true for a finite measure? Namely, let a finite (probability) measure $\mu(\cdot)$. Does a point-wise ...
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3answers
122 views

measurable functions. Why defined like this?

Let us have measurable spaces $(S_1, \Sigma_1)$ and $(S_2, \Sigma_2)$. Idea of measurable function $f$ with respect to $\Sigma_1,\Sigma_2$ is the following. $f:$ $S_1 \to S_2$ has to be such that: ...
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1answer
18 views

Evaluating a contour-integral.

Consider the ellipse $C$ given by $x^2 + y^2/4 = 1$. How to evaluate $$\int_C x^2 \, \nu(d(x,y))$$ where $\nu$ is the Lebesgue length measure on $C$? I am not sure if this can be computed like a ...
2
votes
1answer
63 views

Operator Norm of the Multiplication Operator

Let $(\Omega, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $g$ a measureable function on $\Omega$. Fix $p\in[1,\infty]$ and consider the multiplication operator $$M_g:L^p(\Omega, ...
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0answers
32 views

Measurability of the Jordan decomposition

Let $M := M[0,1]$ denote the space of finite signed Borel measures on $[0,1]$ (the dual of the Banach space of continuous functions $C := C[0,1]$ equipped with the supremum norm) and $M^+ \subseteq M$ ...
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3answers
508 views

Probability and measure theory

I'd like to have a correct general understanding of the importance of measure theory in probability theory. For now, it seems like mathematicians work with the notion of probability measure and prove ...
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0answers
29 views

Convergence in law and continuity of limit implies uniform convergence of distribution functions

Let $X_n$ be a series of random variables converging in law to $X$. Let $F_n,F$ be the corresponding distribution functions, then if the variable $X$ is continuous (takes every value with probability ...
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0answers
50 views

Null sets as unions of countable sets

Let $\mathcal{C}$ be a collection of countable subsets of $\mathbb{R}$ and let $Z:=\bigcup\mathcal{C}$ be the union of its members. For which (uncountable) $\mathcal{C}$ does $Z$ have Lebesgue ...