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

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26 views

Reference request: Measure theory books using $\omega(\alpha) = |\{f>\alpha\}|$

I am working from Wheeden and Zygmund's Measure and Integral, and they prove theorems such as $\int_E f = -\int_{-\infty}^{+\infty} \alpha d\omega(\alpha)$ where $\omega(a) = |\{x: f(x)>\alpha\}|$ ...
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1answer
38 views

Sigma field generated by a Boole algebra and the monotone class theorem

If $\mathcal{B}$ is a boole algebra on a set $X$ then \begin{equation} \sigma(\mathcal{B}) = \{\cup_{n\in \mathbb{N}} \cap_{m\in \mathbb{N}} A_{nm}, \forall n,m \in \mathbb{N}, A_{nm}\in \mathcal{B} ...
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0answers
27 views

Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”?

Suppose that $A$ and $B$ are $\mu$-measurable sets with $\mu(A\cup B) < \infty$. What additional conditions (on $A$ and $B$ and/or $\mu$) are necessary for the implication $$\mathrm{card}(A) = ...
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1answer
16 views

Under which conditions on $f$ do we've got $\sigma(X)=\sigma(f\circ X)$?

Let $(\Omega,\mathcal{A})$, $(\Omega',\mathcal{A}')$ and $(\Omega'',\mathcal{A}'')$ be measurable spaces $X:\Omega\to\Omega'$ be $\mathcal{A}$-$\mathcal{A}'$-measurable $f:\Omega'\to\Omega''$ be ...
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1answer
29 views

If $X:\Omega\to\Omega'$ and $f:\Omega'\to\Omega''$ are measurable and $f$ is injective, then $\sigma(X)=\sigma(f\circ X)$

Let $(\Omega,\mathcal{A})$, $(\Omega',\mathcal{A}')$ and $(\Omega'',\mathcal{A}'')$ be measurable spaces $X:\Omega\to\Omega'$ be $\mathcal{A}$-$\mathcal{A}'$-measurable $f:\Omega'\to\Omega''$ be ...
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2answers
33 views

Stable set by intersection and by finite union

i'm reading a classical book in measure theory, and there is something i don't get. I would say it is a missprint but since the book is famous, probably there is something i don't get, and i need some ...
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2answers
23 views

an existence question from topological groups

$G$ is a topological group, $A$ and $B$ are the subsets of $G$, we denote $AB$=$\{ab:a \in A, b\in B \}$. Let $G$ be a locally compact Hausdorff topological group, $m$ is a left Haar measure on $G$, ...
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1answer
50 views

Lebesgue measures defined on subspaces of $\Bbb R^n$

For any subspace $V$ of $\Bbb R^n$, we have a special measure $\lambda_V$ which can be described in various ways: Haar measure on $V$, or the measure induced by the metric $V$ inherits from $\Bbb ...
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0answers
31 views

Measure of preimage of nullset under a finite-to-one smooth function is nullset

How can I show rigorously the following which is intuitively clear: Given a smooth function $f: \mathbb R \to \mathbb R$ that is finite-to-one(or which satisfy some other from of 'weak injectivity') ...
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1answer
20 views

Subadditivity of total variation of signed measure

I am trying to understand the proof of subadditivity for the total variation of a signed measure but I can't understand a step of the proof. Let $E=\bigcup_{j=1}^{\infty} E_j$. We want to show that ...
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1answer
31 views

Is a random variable constant iff it is trivial sigma-algebra-measurable?

I found a proof here for a measurable function (instead of probability theory's random variable) being constant if and only if the sigma-algebra generated by it is the trivia sigma-algebra, I think ...
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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 ...
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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 ...
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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
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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 ...
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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 $X$ ...
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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
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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 ...
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0answers
17 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 $x\in \Sigma$ $\Sigma$ ...
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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 ...
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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
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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 ...
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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
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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, ...
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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"?
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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
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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
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2answers
46 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
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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1answer
59 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 ...
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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
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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
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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 $(f_n)_{n \geq 1}$ ...
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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
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1answer
54 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: ...
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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
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2answers
199 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 $0$ ...
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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; ...
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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 ...
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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,
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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
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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
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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 $\mu(A)$? ...
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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
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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 $(f_n)_{n \geq 1},f:E \to \overline{R}$ ...