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

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2
votes
3answers
48 views

Continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $

Give an example of a continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $ If $f \in L^1([a,b]), a< b$ that would mean that ...
0
votes
0answers
19 views

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1$

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1.$ I'm aware this post exists elsewhere, say, here but what I don't understand is why we ...
0
votes
1answer
37 views

Proving that $f(x)=\frac{1}{x^2 \ln x} $ is Lebesgue measurable on $(2, + \infty)$

I have that a set $E$ is Lebesgue measurable if the outer measure: $$\mu^*(E)=\inf_{I_1,...,I_n} \mu (I), E \subseteq I_1 \cup I_2 ,...\cup I_n , I_i-\text{intervals}$$ satisfy the three properties ...
4
votes
1answer
32 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
1
vote
0answers
30 views

Symmetric difference and approximation of measure [duplicate]

Let $\scr{A}$ be an algebra of subsets. Let $(\Omega, \sigma(\mathscr{A}), P)$ be a probability space. Then for each $B \in \sigma(\scr{A})$ and $\epsilon > 0$, there exists $A \in \scr{A}$ such ...
0
votes
2answers
46 views

Solving these types of integrals, using Monotone convergence theorem and Dominated convergence theorem.

I'm allowed to use these two theories and obviously the standard techniques when solving integrals. $$\lim_{n\to \infty } \int_{0}^{1}\frac{n^{\frac{1}{2}} x \ln x}{1+n^2x^2}dx$$ I did a similar ...
0
votes
0answers
23 views

Compactness of the outer measure

I have maybe a naive question about compactness of outer measure (or completion). Let $(E,\mathcal B(E))$ be a Polish space, and $\mathcal M_b(E)$ the bounded Radon measure on $E$. Assume that a ...
0
votes
1answer
22 views

Show: $\int_{\Omega} f d\mu =\int_{]0,\infty[} \mu(E_t) d\lambda_1(t)$

I have troubles understanding one step in the solution for this task: Let $(\Omega,\mathcal{A},\mu)$ be a $\sigma$-finite measure and $f:\Omega \rightarrow [0,\infty]$ be a measurable function. Let ...
4
votes
1answer
47 views

uniform boundedness principle for $L^{1}$

i read this theorem from V.I.Bogachev vol 1 Measure Theory. A family $\mathcal{F}\subset L_{1}(\mu)$,where the measure $\mu$ takes values in $[0,+\infty]$, is norm bounded in $L_{1}(\mu)$ precisely ...
0
votes
0answers
30 views

$\mathbb{R}=\cup_{i=1}^{\infty}A_{i}$, all $A_{i}$ Borel subsets of $\mathbb{R}$ and only contain a finite number of rational numbers?

Can $\mathbb{R}$ be written as a countable union of set $A_{i}$ such that all $A_{i}$ are Borel subsets of $\mathbb{R}$ and only contain a finite number of rational numbers? Moreover, are the ...
0
votes
1answer
20 views

Integrate step function and characteristic function

Let be $f_{k} = \frac{1}{k} \mathbb{1}_{[-k,k]}$ and $f_k: \mathbb{R} \rightarrow \mathbb{R}$. How i can show that $f_{k}$ is integrable $\forall k\in \mathbb{N}$? and HOW to compute this? $f_{k}$ ...
1
vote
1answer
39 views

$\sigma$-algebra generated by random variable : Show that if $\sigma(X)=\sigma(Y)$ then $\sigma(X+Y)\subseteq \sigma(X)$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ be a random variable. The $\sigma$-algebra generated by $X$ is defined as $$\sigma(X):=\{X^{-1}(B)\; | \; B\in B_{\mathbb{R}}\}$$ where ...
2
votes
0answers
63 views

Simple exercise measure theory/$\sigma$-algebras

Is this right? Q: Find an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$, is closed under the formation of countable unions, and is closed under the formation of countable ...
1
vote
0answers
21 views

Complex measures, Real Analysis Folland Problem 3.3.19

Relevant background information: We say that two signed measures $\mu$ and $\nu$ on $(X,M)$ are mutually singular if there exists $E,F\in M$ such that $E\cap F = \emptyset$, $E\cup F = X$, $E$ is ...
0
votes
3answers
31 views

Generated sigma algebra and its countable subcollection [duplicate]

Let $\scr{C}$ be a collection of subsets. Prove that if $A \in \sigma(\scr{C})$ (sigma algebra generated by $\scr{C}$), then there exists a countable subcollection $\scr{C}_A$ of $\scr{C}$ such that ...
2
votes
1answer
22 views

Interchanging finite union of finite intersection of sets

I would like to exchange the set operation : $$\cup_{i=1}^m\cap_{j=1}^{n_i} A_{i,j}$$ to be $$\cap \cup A_{i,j}$$ but it is a bit confusing to keep track of the index, and deduce the formula. Compare ...
2
votes
1answer
43 views

about a product of random variables that converges weakly

Let $(\Omega,\mathcal{F},P)$ be a probability space. Suppose $f_n,g_n, n\in \mathbb{N}$ are sequences of functions on this space such that their product $f_ng_n$ converges weakly in $L^2$ to $h$, say. ...
1
vote
3answers
42 views

What can go wrong if we let sigma algebra to admit the union of uncountable union of elements?

By definition we only allow the union of countable infinite of elements to be also include the $\sigma$ field, why not uncountable many? Is there a historical view behind this?
3
votes
1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...
1
vote
0answers
32 views

If $D_1\subseteq D_2$ then the generated sigma-algebras are such that $\sigma(D_1)\subseteq \sigma(D_2)$

Let $A$ be a set of subsets of $\Omega$. Then, the $\sigma$-algebra generated by $A$ is defined as $$\sigma(A):= \cap \{\mathcal{F}\; |\; \mathcal{F}\text{ is a }\sigma\text{-algebra and }A\subseteq ...
0
votes
0answers
32 views

The volume of null-set

I saw this claim: if $v(E)$ = 0, then E is a null-set. The converse statement is wrong. where $v(E)$ is the Jordan measure of $E$, I am not sure if this right, because I know $\int_E1_E=0 ...
1
vote
1answer
31 views

Bhattacharya Distance on Distributions (Matrices) with Different Number of Variables (Dimensions)

We have two matrices, $A$ and $B$, representing two different probability distributions, with dimensions, $m*n$ and $k*n$, respectively. How can we calculate the Bhattacharya distance or another ...
3
votes
1answer
60 views

Will the conditional expectation always have this “property”?(understanding/explanation of conditional expectation)

Lets say you have a probability space $(\Omega, \mathcal{A},P)$, and a random variable $X: \Omega \rightarrow \mathbb{R}$ on this space. Assume that we have a sub-sigma algebra $\mathcal{G}\subset ...
0
votes
1answer
31 views

written set of functions as a union of Borel measurable set

Denote by $\mathcal{H}$ the set of bounded and continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. I wonder if you can write $\mathcal{H}$ as (not trivial) $F_{\sigma}$ set in ...
2
votes
0answers
30 views

Expectation of a step function and its extension to bounded above functions

Suppose that $$E_{\mu}[e^{-\alpha T}f(X)]=\int_{\chi}f(x)M_{\mu}(dx)$$ where $f$ is a measurable step function on Borel space $\chi$, $M_{\mu}$ is a nonnegative measure on $\chi$, $T$ ...
0
votes
0answers
34 views

Fubini's theorem for finite dimensional vector space?

In Weil's Basic Number Theory, the author used Fubini's theorem to show that finite dimensional subspace over locally compact field $F$ has measure zero. While I know this result, I don't know how ...
0
votes
3answers
58 views

Prove that $\{ (x,y) \in \mathbb{R}^2 \mid -2 < x - y^2 < 8 \}$ is open.

$$B := \{(x,y) \in \mathbb{R}^2 \mid -2 < x-y^² < 8 \}$$ I have to show that this set is in $\mathcal{B}(\mathbb{R}^2)$ (Borel set). I think it's obvious that $B$ is an open set, so I guess ...
1
vote
2answers
50 views

Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$

Assume $b>0$. Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$. We know if $b>1$ the integrand can be bounded and it's just an ...
1
vote
1answer
27 views

Limit of Lebesgue integral over an increment of a function

I am currently reading a proof which uses the following fact which I do not know how to show: For any function $f \in L^1 (\mathbb{R})$, $$ \lim_{ h \to 0} \int_{\mathbb{R}} |f(x+h) - f(x) | ...
0
votes
1answer
26 views

Sigma algebra examples

So the definition is: Let $X$ be a set. A sigma algebra is a collection $\Sigma \subset 2^X$ s.t. (i) $\emptyset \in \Sigma$ (ii) $E \in \Sigma \implies E^c \in \Sigma$ (iii) $E_1, E_2,... \in ...
0
votes
1answer
27 views

In the theorem of coincidence of $R$ and $L$ integrals: are we assuming the $\Bbb L$ $\sigma$-algebra? If not, how to prove the measurability of $f$?

Theorem: If $f: [a,b] \to \Bbb R$ is Riemann-integrable, then $f$ is Lebesgue-integrable, and: $$ \int_a^b f(x) dx = \int_{[a,b]} fd\lambda$$ Here is the proof: (assuming that we know ...
1
vote
0answers
54 views

How to prove that a $f(x)=x^2$ is a Borel function?

How to prove that a $f(x)=x^2$ is a Borel function? I don't know how to do this type of prove. I find the the measure theory very abstract and have some problems with grasping the concepts ...
9
votes
5answers
133 views

The sum of infinitely many $c$s is $c$ implies $c = 0$.

This seems like an obvious claim, but I would like to be able to prove this rigorously. Suppose I have $c \in \mathbb{R}$ satisfying $$\lim_{n \to \infty}\sum_{i=1}^{n}c = c\text{.}$$ How does it ...
1
vote
0answers
39 views

Will the joint probability density exist in this case?

Assume that we have a probability space $(\Omega, \mathcal{A}, P)$, and we have two random variables $X,Y: \Omega \rightarrow \mathbb{R}$. On this space. We can define two measures ...
1
vote
1answer
25 views

How to proof that $f^{-1}(\sigma(\mathcal C))\subseteq\sigma(f^{-1}(\mathcal C))$?

Let $f:X\to Y$ be a function, $\mathcal C$ be a family of subsets of $Y$. I am convinced that $f^{-1}(\sigma(\mathcal C))=\sigma(f^{-1}(\mathcal C))$, where $\sigma(\mathcal A)$ is the ...
5
votes
2answers
53 views

Prove that there exists a sequence of continuous functions $f_n(x)$ such that $f_n \rightarrow f$ pointwise on this interval.

Suppose that the real-valued function $f(x)$ is nondecreasing on the interval $[0,1]$. Prove that there exists a sequence of continuous functions $f_n(x)$ such that $f_n \rightarrow f$ pointwise on ...
0
votes
1answer
25 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
1
vote
1answer
22 views

Is this a sufficient condition for a.e. convergence?

Suppose one has a sequence $(f_{n})_{n \in \mathbf{N}}$ of real-valued, non-negative functions defined on a finite measure space $(X, \mu)$, with the following property: For every $n \in ...
0
votes
1answer
13 views

Let $\mathcal{F}$ be the set of all subsets $A\subset \mathbb{R}$ so that $A$ or $A^c$ are countable. Show that $\mathcal{F}$ is a $\sigma$-field.

Let $\mathcal{F}$ be the set of all subsets $A\subset \mathbb{R}$ so that $A$ or $A^c$ are countable. Show that $\mathcal{F}$ is a $\sigma$-field. Following Durrett's definition in his textbook, its ...
0
votes
1answer
14 views

Generate a Dynkin System from the events

How can I generate a Dynkin System from the events $E_2$ and $E_3$ (if $2$ dice are rolled); $E_1=\{\text{first die shows}\ 1\}$ $E_2=\{\text{second die shows}\ 1\}$ $E_3=\{\text{sum of ...
0
votes
1answer
25 views

How many sets are created by repeatedly intersecting a family of sets?

I have a finite set $X$ and a finite family of subsets $X_i \subset X$, $0 <= i < n$, $n \in \mathbb{N}$. What can we say about the size of the transitive hull of this family with regards to ...
0
votes
1answer
23 views

Distribution of a transformation of normally distributed independent variables.

If $W = \frac{X + YZ}{\sqrt{1 + Z^2}}$ where all variables involved are standard-normally distributed and independent, what is the distribution of $W$? The solution I am reading begins with $$P(W \le ...
4
votes
2answers
35 views

Prove f is not measurable

Let $E$ be a non-measurable set contained in $(0,1)$. we will define $f(x) =x\textbf{1}_{E}(x) + x^3\textbf{1}_{E^C}(x)$ where $\textbf{1}_{E}(x)$ is the indicator function for the set $E$. Does ...
0
votes
1answer
26 views

How to use Fubini's theorem to calculate Lebesgue measure?

I'm having some trouble with this problem. Let n $\geq$ 1 and n is an integer. Use Fubini's theorem to calculate the $n$-th Lebesgue measure of this set: $$\left\{ (x_1, x_2, \dots, x_n) \in \mathbb ...
0
votes
1answer
22 views

identify the interval $[0, 1]$ with the Lebesgue measure to the probability space for tossing a fair coin

I want to identify the interval $[0, 1]$ with the Lebesgue measure to the probability space for infinite tossing a fair coin. I know we can define a probability on cylinder sets (the sets can be ...
1
vote
3answers
61 views

If $\int f d\mu = 1$ with $\mu$ probability measure, then $f(x)=1$ for $f\mu$-a.e. $x$?

Let $f \colon X \to \mathbb R^+$ be a non negative, measurable function (w.r.t. a probability measure $\mu$ on $X$), with $\int f d\mu = 1$. From this it does not follow $f = 1$ $\mu$-a.e. However, ...
1
vote
0answers
18 views

Criterion for a signed measure to be positive

I am reading a proof in which I do not understand the following claim: Let $K$ be a compact metric space and let $M(K)$ be the set of signed Borel measures on $K$. Then the set $P(K)$ of positive ...
0
votes
1answer
23 views

If $E,F \in \mathcal L \implies E \bigcap F \in \mathcal L $ Caratheodory condition of measurability.

I will stop at the part I do not understand. Help is needed! Caratheodory condition: For set $E\subset \mathbb R^n$ we will say that it satisfies the Caratheodory condition ili test measurability if ...
0
votes
1answer
34 views

$L^{\infty}(μ)=(L^{1}(μ))^*$ ( Dual space of $L^{1}(μ)$ )

Let $a,b\in X$ . define $μ(\{a\})=1$ , $μ(\{b\})=μ(X)=\infty$ and $μ(\varnothing)=0$ I don't know the below statement is true or not ! $L^{\infty}(μ)=(L^{1}(μ))^*$ ( dual space of $L^{1}(μ)$ ) ...
1
vote
0answers
30 views

Inclusion of $L^p$ and weak $L^p$ spaces

Let $0<p_0<p_1<\infty$, $0<\theta<1$, and $1/p_\theta=(1-\theta)/p_0+\theta/p_1$. Show that $$L^{p_\theta,\infty}(X)\subset L^{p_0}(X)+L^{p_1}(X).$$ Suppose that $f\in ...