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

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3
votes
1answer
52 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
38 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
65 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
21 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
195 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
421 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
22 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
21 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
38 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
40 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 ...
1
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 $\nu$ ...
0
votes
0answers
20 views

Definition of Strongly Absolutely Continuous

I recall seeing a definition of the phrase Strongly Absolutely Continuous in measure theory, but I can't seem to find it anywhere now. Can anyone provide a definition, or am I misremembering?
3
votes
1answer
49 views

Is Riesz measure an extension of product measure?

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
0
votes
1answer
51 views

$\lim_{y \rightarrow^{nt}x}\int \omega(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$

We need to show $\lim_{y \rightarrow^{nt}x}\int \omega_j(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$ Here is the necessary information ...
1
vote
1answer
31 views

Obtaining the measure of a set from a limit of measures of open sets

Let $S$ be a Polish space and $\mathcal B$ the Borel $\sigma$-Algebra on $S$. Let $\mu$ be some finite measure on $(S,\mathcal B)$ and $A\in \mathcal B$ such that $\mu (A)>0$. Is the following line ...
0
votes
1answer
15 views

Proving lebesgue increasing convergence thm…

Im not curious about the proof itself. Just want to know why For arbitrary $c<\int f d\lambda $, if $I>c \rightarrow I \geq \int f d\lambda$ I have seen this kind of argument in some of proof ...
7
votes
2answers
261 views

Does $f\Big(x+\frac{1}{n}\Big) \to f(x)$ for a.e. x as $n \to \infty$?

Let $f$ be a bounded measurable function on the real line, then is it true that $f\Big(x+\frac{1}{n}\Big) \to f(x)$ for a.e. $x$ as $n \to \infty$ I found a result where this is true for $f\in ...
1
vote
0answers
10 views

What is the meaning of “mass defect” in measure theory?

What does the term "mass defect" mean in measure theory? I stumbled upon it in the context of weak convergence and the dominated convergence theorem, but I haven't seen it defined anywhere.
1
vote
2answers
72 views

$\int^b_a t^kf(t) dt\,=0$ for all $k \geq 1 \implies f=0$ a.e.

Let $f\in L^1[a,b]$ satisfying $$\int^b_a t^kf(t) dt\,=0$$ for all positive integer $k$. Show that $f=0$ a.e. I did a similar problem where $\int^b_a t^kf(t) dt\,=0$ was true for all $k\in ...
2
votes
0answers
33 views

Is there a measure theoretic version of Stokes's theorem?

Is there a way to generalize Stokes's theorem on manifolds to general measure spaces? This idea came from trying to generalize the fundamental theorem of calculus to general function/infinite ...
1
vote
2answers
42 views

Real analysis: Show the limit $\lim nμ(E_n) = 0$.

Let $(X, R, μ)$ be a measurable space. Let $f$ be a measurable and integrable function on $X$. Let $E_n = \{x ∈ X|f(x) > n\}$. Prove that $\lim nμ(E_n) = 0$. I know that $\mu(E_n)$ is equal to its ...
1
vote
0answers
20 views

Prove the Pre-image of $T(x,y) = (x-y,y)$ maps Measurable Sets to Measurable Sets

Consider the linear transformation $T(x,y) = (x-y,y)$ on $\mathbb{R^{2n}}$. Prove that that the pre-image of $T$ maps measurable sets to measurable sets. Its clear that T is linear, invertible, and ...
4
votes
1answer
18 views

A question about projections of product measure space

I am considering the space $\mathbb{R}^{\mathbb{N}}$ of real-valued sequences with the sigma-algebra $\mathcal{F}$ generated by sets of the form $$\{\omega \in \mathbb{R}^{\mathbb{N}} : \omega_k \in ...
1
vote
0answers
39 views

Intuition behind measurable random variables and $\sigma$-algebra

I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture. I'll try to ...
4
votes
2answers
118 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
0
votes
2answers
46 views

The measure of set

Is it exist a closed set $E$ that is subset of $[a,b]$, $E \neq [a,b]$, which measure is $b-a$? I think, it exists, but can't find an example.