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

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2
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0answers
34 views

Approximating the distribution of the infinte sum of random variables

What is the formal way to show that for an infinite sum of random varaibles $\sum_{i=1}^\infty X_i$, we have $\forall\varepsilon>0,\exists N<\infty$ such that $$0<P\left(\sum_{i=1}^N X_i\leq ...
0
votes
1answer
23 views

Need of proper concept of inverse function in sets

A function $X ∶ (\Omega_1, \{ \Omega_1 , \varnothing\}) \to (\Omega_2 , \{\Omega_2,A,A^c,\varnothing\})$ is given and $A$ is some non empty subset of $\Omega_2$. Now since I am new to measure theory a ...
1
vote
0answers
7 views

Measure and Probability Malcolm Adams [duplicate]

I was try to find the solution of two exercises but I don't have any ideas,can someone help me? The first one is: prove a refinement of the strong law of large numbers ,which says that $S_N/N^δ →0 ...
3
votes
1answer
35 views

Function is measurable if and only if it is constant a.e.

Let $(X,\Sigma,\mu)$ be a measurable space with $\mu(X)=1$ and $\mu(A) \in \{0,1\}$ for all $A \in \Sigma$. Show that $f: X \to \mathbb R$ is $\mu-$ measurable if and only if $f$ is constant a.e.. I ...
0
votes
1answer
25 views

Prove a set obtained from sequence of measurable sets is measurable

Exercise Let $(X,\Sigma,\mu)$ be a measure space and let $(A_k)_{k \in \mathbb N}$ be a sequence of measurable sets. For each $m \in \mathbb N$, we define $B_m$ as the subset of all points in $X$ ...
3
votes
2answers
39 views

Lebesgue integrable function, convergent series

I am trying to solve the following: Let $(X,\Sigma, \mu)$ be a measurable space, $f:X \to \mathbb R$ measurable and let $A\in \Sigma$. For each $n$ natural number, we define $A_n=\{x \in A: |f(x)| ...
2
votes
1answer
24 views

Integrable function $f$ on $(\mathbb N, \mathcal P(\mathbb N),\mu)$ and series

Problem Let $(\mathbb N, \mathcal P(\mathbb N),\mu)$ where $\mu(A)=card(A)$. Show that $f \in L^1(\mathbb N,\mu)$ if and only if $\sum_{n=1}^{\infty} |f(n)|<\infty$, in which case $\int_X f ...
0
votes
0answers
16 views

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$,

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$, $\forall g \in L^2({\sigma})$ here $x\in \Sigma$ $\Sigma$ ...
0
votes
2answers
32 views

Calculate limit of Lebesgue integrals

I am trying to calculate this limit: $$\lim_n \int_0^{n^2}e^{-x^2}\sin(\frac{x}{n})dx$$ Since $$\int_0^{n^2}e^{-x^2}\sin(\frac{x}{n})dx=\int_{[0,\infty)}e^{-x^2}\sin(\frac{x}{n})\mathcal ...
0
votes
0answers
23 views

Law of iterated logarithm for Markov Chains

Does anyone know where(or if) I can find a proof of law of iterated logarithm for irreducible and aperiodic Markov chain with finite number of states. All of the proofs I have seen so far are really ...
3
votes
1answer
18 views

Proof of uniqueness of conditional expectation

I have a question on the proof Durrett (p. $190$) gives for the uniqueness of the conditional expectation function. If I understand his proof correctly, here is what I think it is saying: Suppose ...
1
vote
2answers
44 views

Constructing measure preserving maps between non-atomic measures

Suppose $(\mu, X,\Sigma)$ and $(\mu^\prime, X^\prime, \Sigma^\prime)$ are non-atomic probability measures. Is it always possible to construct a measure preserving map between the two spaces? (If ...
1
vote
1answer
17 views

Finding linear orders on a measure space whose initial segments have all possible measures

Let $\mu$ be a non-atomic probability measure on some space $(X, \Sigma)$. Is it always possible to find a linear order, $\leq$, on $X$ such that $\mu: \mathcal{A}\rightarrow [0,1]$ is surjective, ...
0
votes
0answers
26 views

Is an infinite set always equinumerous to either set of natural or real numbers? [duplicate]

Is an infinite set always equinumerous to either set of natural or real numbers? Is there any set "between"? Or maybe "beyond"?
0
votes
1answer
28 views

Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra

I am trying to prove that collection of all finite dimensional cylinder sets is an algebra but not $\sigma$-algebra. Cylinder sets are defined as: $\mathcal{B}_n$ is defined as the smallest $\sigma-$ ...
3
votes
1answer
54 views

Lebesgue integrable function $g$ equals characteristic function

I am trying to solve this problem: Let $g:[0,1] \to \mathbb R$ be a non negative integrable function over $[0,1]$. Prove that if there is $\alpha \in \mathbb R$ such that for all $n \in \mathbb N$, ...
3
votes
2answers
45 views

Continuity vs. Mapping open sets to open sets?

I have a question and I have no idea how to solve this: One problem in my Real Analysis text book says: Show that if $\ell$ is a nonzero linear functional on a normed vector space not necessarily ...
4
votes
2answers
33 views

Decreasing Sequence of Measures

For $(X,\mathcal{F})$ a measure space, I know that if we have $\mu_{n}(A) \searrow$, i.e. is a decreasing sequence of measures for each $A \in \mathcal{F}$ and $\mu_{1}(X) < \infty$ then $\mu = ...
1
vote
1answer
30 views

Finite additive measure

Problem: Let $[0,1]\cap\mathbb{Q} $ denote the set of all rational number inside the interval $\left[0,1\right]$, let $\mathcal{A}$ be the algebra of sets that can be expressed as finite unions of ...
-1
votes
1answer
34 views

Proof of Caratheodory Extension theorem

I was trying to prove that $\mu^*$ is an outer measure. I was easily able to solve the first two conditions of an outer measure(That $\mu^*\ge0 $, and the monotonicity condition) however I have been ...
0
votes
1answer
30 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuos stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
0
votes
1answer
40 views

Is this a measurable function

Let $\Omega_1 = \{ a, b, c, d \}$ and $Ω_2 = \{ 1, 2, 3, 4, 5 \}$ , and assume $F_i = \mathcal P ( \Omega_i ) ,\space i=1,2$. Consider a uniform probability assignment over $\Omega_1$ . For the map ...
1
vote
0answers
39 views

Fundamental theorem of calculus for the Lebesgue integral

Let $\lambda$ be the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ $f:\mathbb{R}\to\mathbb{R}$ be $\lambda$-integrable What's the easiest way to show $$\frac ...
0
votes
1answer
58 views

A multiple of a characteristic function is the weak limit of a sequence of characteristic functions

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...
8
votes
2answers
108 views

Let ${f_n}$ be a sequence of integrable functions on $\mathbb{R}$ such that $f_n\rightarrow f$ almost everywhere.

Let ${f_n}$ be a sequence of integrable functions on $\mathbb{R}$ such that $f_n\rightarrow f$ almost everywhere. We also have $f\in L^1(\mathbb{R})$ and $\int_{\mathbb{R}}f_n\rightarrow ...
1
vote
2answers
66 views

The length of a point and the interval

I think the length of a point is $0$, and since biunique corespondence between the points of [0, 1] and [0, 10], therefore I came to the conclusion that there is a same number of points between [0, 1] ...
2
votes
1answer
34 views

Sequence of measurable functions $f_n=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$, uniform convergence

For each $n \in \mathbb N$, let $f_n:[0,\infty) \to \mathbb R: f_n(x)=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$. Show that there is no $E \subset [0,\infty)$ such that $|E|=0$ and $(f_n)_{n \geq 1}$ ...
1
vote
0answers
25 views

Sequence of measurable functions on finite measurable set

I am struggling to solve the following exercise: Let $E \subset \mathbb R^d$ be finite measurable and $(f_k)_{k \geq 1}:E \to \mathbb R$ be a sequence of measurable functions such that for all $x \in ...
5
votes
1answer
53 views

Decay of Fourier Transform

I encountered the following statement, and I cannot see why it is true(if it is). Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: ...
0
votes
1answer
22 views

function of bounded variation and properties

I have to prove that if $f : [a,b] \rightarrow \mathbb{R}$, $g : [a,b] \rightarrow \mathbb{R}$ are of bounded variation so it is $f \cdot g$. I want to use the definition to prove this but I don't ...
5
votes
2answers
198 views

Determining a measure through a class of measure preserving functions

Let $\mu$ and $\mu^\prime$ be probability measures over the sigma algebra $\Sigma$ consisting of the Lebesgue measurable subsets of $[0,1]$. Suppose also that $\mu$ and $\mu^\prime$ assign measure $0$ ...
0
votes
0answers
2 views

inferring parameters from limting relative frequencies

I refer to my previous question concerning what i call the converse strong law of large numbers (instead o the normal SLLN given the probability=p that with prob1, the limiting relative frequency=p; ...
0
votes
1answer
26 views

$\lambda$-system

$\Omega = \{a,b,c\}$ $\mathcal{C}=\{\{a\},\{b\}\} \subset \mathcal{P}(\Omega)$ What is the $\sigma$-algebra and the $\lambda$-system generated by the class $\mathcal{C}$ described above? Will the ...
5
votes
3answers
425 views

Is the empty set Lebesgue measurable?

I have a quite dumb question. Is the empty set measurable? say with respect to the standard measure. I totally acknowledge intuitive explanations. Thanks,
0
votes
2answers
23 views

Compute the outer measure of $1+ \frac{1}{n}$

Let us have a fixed interval $I_0=[a,b]$ and let $A$ be a subset of $I_0.$ Compute $$\mu^* \left( \left\{\left( 1+ \frac{1}{n}\right)^n | n \in \mathbb{N} \right\} \right)$$ I've been thinking that ...
0
votes
1answer
22 views

$\mu^* \left( \bigcup_{n=1}^{\infty} A_n\right) = 0$

Let us have a fixed interval $I_0=[a,b]$ and let $A$ be a subset of $I_0.$ Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of subsets of $I_0$ s.t $\mu^* (A_n)$ (outer measure) is 0 for all natural $n$. ...
3
votes
1answer
35 views

Lebesgue measure. Find $\mu(A)$

If $I_0 = [a,b]$ and $b>a$, let $A \subset I_0$ be a measurable set such that $$\forall p,q \in \mathbb{Q} , p \neq q \rightarrow (\{p\}+A)\cap(\{q\}+A) = \emptyset$$ Then what is $\mu(A)$? ...
1
vote
1answer
37 views

Outer measure > $0$?

Let's say we have $A \subset I_0$ as an arbitrary set such that $Int(A) \neq \emptyset$ My question is: is $\mu^* (A)$ always non-negative/positive?
2
votes
1answer
33 views

Application of Egorov's theorem

Problem Let $(E,\Sigma, \mu)$ be a $\sigma$-finite measurable space (i.e., $E=\bigcup_{k \in \mathbb N} A_k$ where $\mu(A_k) < \infty$ for each $k$). Let $(f_n)_{n \geq 1},f:E \to \overline{R}$ ...
1
vote
2answers
39 views

Measure of intersection of set and its translation

I came across an old qualifying exam question: Let $A\subset [0,1)$ be a Lebesgue measurable subset of unit intreval such that $0<\mu(A)<1$. For every $x\in [0,1)$ let $A+x=\{x+y$ mod 1$:y\in ...
3
votes
2answers
41 views

Measure theory problem to show a set contains positive interval [duplicate]

Let $E\subset \mathbb{R}$ be a Lebesgue measurable subset of reals such that $\mu(E)>0.$ Consider the set $E+E=\{x+y: x,y\in E\},$ prove that $E+E$ contains an interval of length greater than $0$. ...
0
votes
1answer
24 views

Outer measure proof for rational numbers

I saw this problem solved for particular cases like $(0,1)$ but never for general. If $A \subset \mathbb{Q} \cap (a,b)$ and $a<b$ (set of all rational numbers in $(a,b)$) Claim: For every ...
6
votes
2answers
70 views

Looking for a “job description” for Hölder's inequality

Here's an example of what I mean by "job description" in the post's title: triangle inequality: to be used, whenever the (unsigned) distances between adjacent points in a sequence $x_0, x_1, x_2, ...
4
votes
0answers
101 views

Sigma algebra generated by a homeomorphic random variable

Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every ...
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vote
1answer
24 views

if $|f_n|<g \in L^1$, and $f_n \rightarrow f$ in measure, how do we know $\lim_{n\to \infty} \int f_n = \int f$

I know that a subsequence converges, but I am not even convinced that $\int f_n$ converges at all. They are all finite, but I am not certain how to bound them. I have considered working with $\int ...
2
votes
2answers
33 views

Verify that the set $\Omega = \lbrace (u,v) \in \mathbb{R}^2 \mid |u| + |v| \leq 1 \rbrace$ is Jordan measurable

Motivation: I am currently in a rather uncomfortable spot in my Analysis studies. In class we introduced the Jordan measure in a very vague way, meaning no proofs, no examples. (Because next Semester ...
0
votes
1answer
38 views

Definition of Lebesgue measure on the Circle $X=\mathbb{R}/\mathbb{Z}$.

I was given the following definition: I feel this is not adequate and would only define Lebesgue on $[0,1]$? (Wikipedia uses a pushforard of the complex exponential) Also later I am given a proof ...
1
vote
1answer
46 views

How to show that $C$ is countable?

Let $C=\{B_a:a\in A\}$ be a collection of piecewise disjoint measurable subsets of $[0,1]$ having positive Lebesgue measure. How to show that $C$ is countable?
0
votes
1answer
31 views

$\lim_{k \rightarrow \infty} \mu \left(\bigcup_{n=1}^k A_n \right) = \mu \left(\bigcup_{n=1}^\infty A_n \right)$

The problem I'm working on is: Prove that for a family of measurable sets $A_k$ in $[a,b]$ the following is true $$\lim_{k \rightarrow \infty} \mu \left(\bigcup_{n=1}^k A_n \right) = \mu ...
2
votes
0answers
16 views

Blow-up of derivative of BV function at the jump set

"Motivation" Let $u\in BV(\mathbb{R}^n)$ be a function of bounded variation, and let $x\in J_u$ be a point in its jump set. For $\mathcal{H}^{n-1}$-a.e. such $x$, we can define the unit normal $\nu$ ...