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

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Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
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195 views

Another version of Egoroff's theorem

Let $(X,M,\mu)$ be a finite or $\sigma$-finite measure space. Let $\{f_n\}$ be a sequence of finite a.e., measurable functions such that $f_n \to f$ [a.e.]. Then there is a partition of $X$ into a ...
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111 views

Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
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329 views

weak convergence of independent sequence

I want to show that if $X_n\to^w X$ and $Y_n\to^w Y$ which is 'weak convergence' and the $X_n,Y_n$ are independent RV's on the same probability space, Then we also have weak convergence of the ...
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234 views

Alternative version of monotone convergence theorem

Let $(X,A,\mu)$ be a measure space and let $(f_n)_{n\in\mathbb N}$ a sequence of positive measurable functions s.t. $$ f_1(x)\ge f_2(x) \ge \ldots \ge f_n(x)\ge ... \ge 0, \qquad \forall x\in X ...
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278 views

A question on semifinite measures

First, I give the definition from Folland, Definition: Let $(X, \mathcal{M}, \mu)$ be a measure space. If for each $E \in \mathcal{M}$ with $\mu(E) = \infty$, there exists $F \in \mathcal{M}$ ...
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166 views

essential supremum of a matrix multiplication operator

Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a ...
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382 views

Limit of a decreasing sequence of measurable sets.

Let $(X,\mathcal{A})$ be a measurable space, with measure $\mu$. Let $\{E_n\}_{n \in \mathbb{N}} \subseteq \mathcal{A}$ be a sequence of measurable sets, with $E_{n+1} \subseteq E_n, \ \forall n \in ...
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115 views

Show a function is in $L_\infty$

Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ...
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253 views

What is the sigma algebra generated by the indicator function of random variable?

I was thinking what is the $\sigma$-algebra generated by the random variable $Z= \mathbb{I}(X+Y=0)$ where $X,Y\sim\operatorname{Bern}(p)$ iid. (Note: $\mathbb{I}$ is the indicator function.)
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81 views

Sequence of continuous fuctions $f_n:[0,1]\rightarrow [0,1]$ s.t. $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ but…

Give an example of a sequence of continuous functions $f_n:[0,1]\rightarrow [0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n(\varepsilon)) = 0$ for every $\varepsilon >0$ but ...
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144 views

Trying to do a measure theory proof concerning almost everywhere convergence

Let $E\subset\mathbb{R}$ be measurable and for each $n\in\mathbb{N}$, let $f_n:E\rightarrow\mathbb{R}$ be measurable. For each $\varepsilon > 0$, define $E_n(\varepsilon) = \{x\in E : |f_n(x)|\geq ...
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73 views

How to bound $L^p$ norm of a product

I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate ...
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44 views

The Fourier transform of $x \mapsto |x|^{-1/3}e^{-x^{2}}$ is not in $L^{1}$.

Okay previously my lecturer showed that this is so by proving in the following way: Proof by contradiction. Suppose the transform is in $L^{1}$. Then as $f \in L^1$, we may use Fourier Inversion ...
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376 views

Pointwise convergence of Lebesgue-integrable functions

Is it true that if $(f_n)_{n\geq 1}$ is a sequence of Lebesgue-integrable functions on $[0,1]$ that converges pointwise almost everywhere to a function $f$, then $f$ is integrable? Edit: If not, then ...
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139 views

Expressing a Countable Union of Sets as the Countable Union of Disjoint Sets (e.g. Eliminating Duplicate Intersections)

Consider a set $E=\bigcup_{j=1}^{\infty}E_{j}$. How can I re-express $E$ in such a way that all of the contributions to $E_{n}$ are covered only once. Presumably, such a re-expression of $E$ will be ...
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100 views

Lebesgue measure: show that $\mu^*(k\,A)=k \mu^*(A)$

For $k\gt 0$ and a subset $A$ of $\mathbb{R}$, let $k\,A=\{kx\mid x∈A\}$. Show that $$\mu^*(k\,A)=k \mu^*(A)$$ and that $A$ is measurable if and only if $k\,A$ is measurable.
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380 views

A continuous function with measurable domain.

Let $D$ and $E$ be measurable sets and $f$ a function with domain $D\cup E$. We proved that $f$ is measurable on $D \cup E$ if and only if its restrictions to $D$ and $D$ are measurable. Is the ...
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455 views

Measure theory - Lebesgue measure

I have two problems that I would like some help with. Show that every countable subset of $\mathbb{R}$ has Lebesgue measure zero. For two arbitrary sets $A$ and $B$ show that $$\lvert ...
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159 views

Show that the difference of two Borel sets is itself a Borel set

I am currently in a Measure Theory class and we are going over Borel sets. I am having difficulty with the following proof: show that for any two Borel sets $A,B$, the difference $A-B$ is a Borel ...
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82 views

Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$

Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$ cX := \{ cx \mid x \in X \}. $$ Then $$ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$ Now I can prove this for ...
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162 views

Bochner integrability

I have a question, which is related to the notion of Bochner integrability. In the course of solving some particular problem, I need to show that \begin{align} \sqrt{\sum_{k=1}^\infty (\lambda_k + ...
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260 views

Expectation function of random variable Lebesgue integral

If $X$ is a random variable and $h$ a Borel measurable function, let $\mathbb{P}^{x}$ be the distribution of $X$. I now have problems proving that $h(X)\in L^1(\Omega,\mathcal F,\mathbb{P})$ if and ...
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166 views

Not countably additive set function

If $\mu: A \longrightarrow \left\{0,\infty\right\}$ is a set function where $A$ is the set of all left open right closed intervals in $(0,1]$ defined as: $$ \mu\bigl((a,b]\bigr) = \left\{ ...
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72 views

Sigma-algebra without exhausting sequence?

Let me first state the definition of an Exhausting Sequence of Sets: Let $X$ be a set, and let $\mathcal{S} \subset \mathcal{P}(X)$ be a collection of subsets of $X$. An exhausting sequence of sets ...
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242 views

Lebesgue integral over subsets

Let $v(E)=\int_{E} f d\mu=\mu (1_{E}f)$. Take $f \in \Sigma^{+}$ a non-negative measurable function and h a measurable function, $h\in \Sigma$. I now need to show that $h \in L^{1}(S,\Sigma,v)$ if ...
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116 views

Defining a probability distribution

I am taking a probability theory course, and I need help understanding a homework question. Let $(\Omega , \mathcal{F}, \mu)$ be a measure space, and $f:\Omega\rightarrow[0,\infty)$ be such that ...
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36 views

Regd. Riemann Rearrangement

Assume that A is an arbitrary set and there exists a bijection $\phi : B \rightarrow A$ and $x_{\alpha} \in [0,\infty]$, the book says that $$\Sigma_{\alpha \in A} x_{\alpha}= \Sigma_{\beta\in B} ...
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90 views

Limit Volume of Parallel Sets

Given $F \subset \mathbb R^n$ non empty and $\epsilon > 0$. Let $F_\epsilon$ be $\epsilon$-parallel set of $F$, $$F_\epsilon := \{x \in \mathbb R^n:d(x,F)\le\epsilon\},$$ with $d(x,F):= \inf_{y\in ...
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74 views

Boole and Chebychev inequalities on $(\Omega,\mathcal{B},P)$

Given a sequence of random variables $X_1,X_2,...$ defined on the same probability space $(\Omega,\mathcal{B},P)$. part 1: Verify that $P(\limsup_{n\to\infty}X_n>x)=0$ if and only if ...
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127 views

Distinguish between Algebras

$$ S_n = \mathscr P (\{ -n, -n+1, \ldots, n-1, n\}) $$ $$ R_n = \{r : \Omega - r \in S_n\} $$ $$ T_n = S_n \cup R_n$$ I need to check whether $T_n$ is an algebra, semi-algebra or sigma algebra. ...
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582 views

Problem in proving a set is a Sigma Algebra

Let $(\Omega,\mathscr A)$ be a measurable space. If $\varnothing \subset X \subset \Omega$, let $$\mathscr F = \{ F \subseteq \Omega, F = X \cap Y, Y \in \mathscr A\} \;. $$ I need to prove that ...
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570 views

Example that a measurable function $f$ on $[1,\infty )$ can be integrable when $\sum _{n=1}^{\infty }\int_{n}^{n+1}f$ diverges.

I am seeking help in my attempt to formulate a proof to disprove the following. For a measurable function $f$ on $[1,\infty )$ which is bounded on bounded sets, define $a_n= \int_{n}^{n+1}f$ for each ...
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192 views

Having trouble with proofs using generated sigma algebras

I've just started with an advanced course in measure theory and I'm having trouble working with $\sigma$-algebras. Here's my problem: Let $(S,\Sigma, \mu)$ be a measure space. Call $N \subset S$ a ...
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Proof of the Monotone Class Theorem

I am learning the Monotone Class Theorem from Jacod's Probability Essentials. I don't quite understand the idea of the proof in the book. I don't see the point in the proof at all. What's the use ...
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274 views

Right continuous function that is continuous over rationals

Let's assume that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a right continuous function that is continuous on rational points. It is clear that f need not to be continuous. In fact it fails to be ...
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655 views

conditional expectation of normal distribution using sigma algebra

Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities. Let $Y = IX$. How would I find the distribution of $Y$ and ...
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106 views

$L_p$ space convergence revised

Here's what I really wanted to ask Suppose that $f_n \rightarrow f$ a.e. If we have $\int{\left|\lvert f_n\rvert^p - \lvert f\rvert^p\right|} \rightarrow 0$, is it true that $\|f_n\|_p \rightarrow ...
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541 views

Uniform integrability question

I know I keep asking similar sorts of questions, but I want to understand exactly what is going on here. So, I will list a full definition and then the problem. (Basically, I want to prove something ...
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111 views

Changing order of integration/expecation, probability problem

Consider the ON-sequence $\{\varphi_{k}\}_{k}\in L^{2}(\mathbb{R})$ and let $$I_{k}\in \mathrm{Bernoulli}(\lambda_{k}),\;\sum_{k=1}^\infty \lambda_{k}<\infty$$ $ \{I_k\} $ are all independent. ...
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89 views

How to “project” a measure

Given a borelian measure in $\mathbb{R}^2$, there is a canonical way or simply a way to obtain a measure on a line, for example $x=0$? (a measure with support in the line I'm considering). The ...
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51 views

For any $c_1,c_2\in\mathbb{C}, E(c_1Z+c_2)=c_1E(Z)+c_2$

When dealing with real-valued RVs, the extensions of expectation and variance are quite clear to me. For example, showing $E(aX+b)=aE(X)+b$ and $var(aX+b)=a^2var(X)$ is relatively straightforward to ...
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185 views

First hitting time of leftcontinuous process

Suppose that we are working with a Filtration which is right continuous. I know then, that the first hitting time of a right continuous process into an open set is a stopping time. Is the same true, ...
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497 views

Are Cumulative Distribution Functions measurable?

It is well-known that CDFs (Cumulative Distribution Functions) of one dimensional random variables are Borel measurable. But does the same apply to CDFs of multi-dimensional random variables (rvecs)? ...
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165 views

Outcome of unmeasurable probability

I consider a standard normal random variable $X$ and a Vitali set $V$. $P(X\in V)$ can not be computed as $V$ is not measurable. Now I consider the outcome of the following experiment $E_S$ : $N_S$ ...
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87 views

Question on whether a set is measurable

Let $A$ be a non-Lebesgue measurable set, and let $B=[0,1]\subseteq\mathbb{R}$. Show that $C=A\times B$ is non-measurable. I try use the regularity of the lebesgue measure, but don´t work, maybe ...
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68 views

Measure spaces such that the semi-norm is a norm

In my lecture notes there is the following exercise: "Characterize those measure spaces $(X, B, \mu)$ on which the semi-norm $\|f\| = \int_X |f| d \mu$ defined on $L^1(X) = \{ f \mid f \text{ ...
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123 views

Explanation of statement needed (Bochner-style integral, Fubini's theorem, etc.)

I am reading a paper. They define $$ L_{p,q}(Q) = \{ u \in L_p((0,T); L_q(Y)) : u(t, \cdot) = 0 \text{ on } Y \backslash Y_t \text{ for a.e. $t \in (0,T)$}\}$$ with norm $$\lVert u ...
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401 views

Sequence of Uniformly Bounded functions

Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$. Assume the following. For any sequence $\{X_k\}_{k=1}^{\infty}$ of ...
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387 views

minimum $ \sigma-$algebra contains intervals with rational endpoints

What is the minimum $ \sigma-$ring and $\sigma-$algebra on $\mathbb R$ which contains the intervals with rational endpoints ? Is there a relation between this $ \sigma-$ algebra and Borel's ?