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

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how to prove that the sequence of random variables converges to a random variable?

If $Z_1,Z_2,...,Z_n$ are random variables such that $Z(\omega)=\lim_{n \to \infty}Z_n(\omega)$ exists $\forall \omega \in \Omega$ then Z is also a random variable I was reading a book on probability ...
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Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
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Borel Measures: Atoms (Summary)

Disclaimer: This is a summary of the discussions: Measure Atoms: Definition? Borel Measures: Discrete & Continuous? Borel Measures: Atoms vs. Point Masses Reference: Further results are ...
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8 views

What is the definiton for “best probability measure”?

I'm looking for this definition is notes that use the phrase and elsewhere, but it just isn't there. Does anyone else recognize the phrase?
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1answer
27 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
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Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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How can I say whether $g \in \bar {\mathcal M}(\mathcal B(\mathbb R))^+$ such that $\int g \ d(\lambda) < \infty$ and $|f_n| \le g$ exist?

Let $f_n(x)=\frac n 2 \cos(x)1_{[\frac {-1} n, \frac 1 n]}(x)$ be a function $\mathbb R \rightarrow \mathbb R$. I've shown that $f_n \in \mathcal L^1(\lambda)$ and $\int f_n \ d(\lambda) = n ...
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$\mu_x\otimes\mu_y$ and $\mu_y\otimes\mu_x$

Let $X$ and $Y$ spaces endowed with measures $\mu_x$ and $\mu_y$ defined on set semirings $\mathfrak{S}_x$ and $\mathfrak{S}_y$ and let $A\subset X\times Y$ be a subset of $X\times Y$ such that ...
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1answer
34 views

Consistency strength of 0-1 valued Borel measures

The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", ...
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1answer
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On Egorove's Theorem

In the book Measure and Intrgral - Wheeden, Zygmund (p.57), I saw the Egorove's theorem and its proof. I puzzled with the statements of the Egorove's theorem, and a Lemma needed in the proof of ...
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46 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
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2answers
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Question about sigma-algebras

Assume some random variables $$X_1,\dots,X_n : \Omega \to \mathbb{R}$$ are given where $(\Omega,\Sigma)$ and $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ denote measurable spaces. How can one proof that $$ ...
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approximation simple functions with finite support

Let $f$ be a nonnegative measurable function. I want to prove that there is an increasing sequence of nonnegative simple functions each of which vanishes outside a set of finite measure such that ...
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1answer
9 views

Measure of a Set in Relation with a Bounded Function

If $f:[0,1]\to\mathbb{R}$ is bounded, then, for a given $\epsilon$, can the set $S:=\{x\in[0,1]:f(x)>\sup f-\epsilon\}$ be of (Lebesgue) measure $0$ ? If $f$ is continuous then I think the answer ...
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1answer
42 views

In frequentism, does every event have a probability?

For an infinitely repeatable trial with event space $\Omega$, and an event $A\subseteq \Omega$, the frequentist probability of $A$ is defined: $P(A):= \lim_{n\rightarrow\infty} \frac{n_a}{n}$, where ...
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1answer
19 views

What does $\mathbb{R}$-invariant mean for a measure?

Let $(X, A, m)$ be a measure space with m being $\mathbb{R}$-invariant. What does this mean?
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Gamma function in $C^{2}$

How can I show that for $x>0$, the Gamma function is at least $C^{2}$? The Gamma function is defined as $$\displaystyle \int^\infty_0 e^{-t}t^{x-1}\ dt$$ For which $x$ is the integrand integrable?
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Lebesgue Measure: No Atoms!

Disclaimer: This is just meant as record of a proof. For more details see: Answer own Question How to prove that the Lebesgue measure has no atoms: $$\lambda:\mathbb{R}^n\to\mathbb{R}_+$$ (Recall ...
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Discrete set that is compact and jordan measurable

We define a) C(0) = [0,1] b) C(n) = New set that is obtained by erasing 1/3^n section long from the middle of the remaining section in C(n-1) *If C(0) 0---------1 Then C(1) = 0---xxx---1 C(2) = ...
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Cofinite Topology: Borel Algebra?

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why ...
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Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\in\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ a ...
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Set $E$ which halves the measure of an open interval [duplicate]

This was an exam question. I know that my answer is wrong, but I believe myself to be on the right track. Can someone help me finish my construction? Here is the question. Find a set $E$ with the ...
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1answer
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If $U \cap \iota(\Bbb{R}^k)$ is a $k$-dimensional null-set for every **linear** embedding $\iota : \Bbb{R}^k \to \Bbb{R}^n$, $U$ has measure zero

In the post Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?, the OP originally posed (essentially) the following question: ...
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1answer
21 views

Show that sigma algebra generated by subsets of R contains sigma borel algebra of R.

So basically I want to show that if (a,b) is in F, then S(F) contains B(R). (b>a) (Where S(F) is the sigma algebra generated by F and B(R) is the Borel sigma algebra generated by the real numbers.) ...
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1answer
20 views

Proof of Outer Regularity of Lebesgue Measure on R

Let $E \subseteq \mathbb{R}$ be a measurable set, and $\epsilon > 0 $. Show that there is an open set $G \supseteq E$ such that $\mu(G \setminus E) < \epsilon$. Any hints? By the definition of ...
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Equivalent Measures via Hahn-Kolmogorov and $\sigma-$finiteness

Hello Mathematics community. I am currently struggling with the following problem from Terry Tao's Introduction to Measure Theory textbook. It deals with pre-measures and the Hahn-Kolmogorov ...
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Does Fubini's theorem imply $\int_X (\int _Y f _x d \lambda )d \mu=\int _X d \mu \int _Yf(x,y) d \lambda$?

I need some help with intepretating the result of Fubini's theorem. define $ \phi (x) =\int_Y f _x d \lambda $ and $ \psi (y)= \int _X f _y d \mu$ According to Rudin, Fubinis theorem tells us that ...
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If $\mu(B)=0$ then $\mu_y(B_x)=0$

Let $B_x$ be the $x$-section of a $\mu_x\otimes \mu_y$-measurable set $B$, where $\mu_x\otimes \mu_y$, which I will call $\mu$, is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ ...
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1answer
28 views

Is a function with everywhere discontinuities of the second kind always measurable?

Let $f : [0,1] \to \left\{ 0, 1 \right\}$ be a function that has at each point a discontinuity of the second kind. Is $f$ measurable if we equip the domain with the Borel or even Lebesgue ...
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Question about Girsanov theorem

Tn the book "Stochastic Differential Equation" from Oksendal one can find the following theorem(6th edition, Theorem 8.6.8): Let $X(t)=X^x(t)$ and $Y(t)=Y^x(t)$ be an Itô diffusion and an Itô ...
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1answer
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Union of $x$-sections measurable?

I know that the $y$-section $A_x$ of a $\mu_x\otimes \mu_y$-measurable set $A$, where $\mu_x\otimes \mu_y$ is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ (both measures being ...
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1answer
12 views

“Lebesgue measure” on metric spaces?

Sry if my question is stupid, but I just wondered if is there is like a corresponding counterpart to the Lebesgue measure on $\mathbb{R}^n$ for (some?) metric spaces $(E,d)$? Since the natural way to ...
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1answer
26 views

Proof for the absolute continuity with respect to Lebesgue measure

Let $\mu$ be a measure. I'm looking for a reference to a proof showing that the following condition is enough to prove absolute continuity with respect to the Lebesgue measure: $$\liminf\limits_{r ...
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I know by Fubinis theorem that $N$ is $\mathcal E$-measurable and $\mu(N)=0$. How can I see that $N \neq X$, that is $N \subset X$?

Let $(X, \mathcal E, \mu)$ and $(Y, \mathcal F, \nu)$ be $\sigma$-finite measure-spaces and consider the product-space $(X \times Y, \mathcal E \oplus \mathcal F, \mu \oplus\nu)$. Let $f: X \times Y ...
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1answer
25 views

Monotone Sequence of Sets

If someone can check my proof for the following statement, would be awesome. Thanks. Suppose $\{A_n\}$ is a monotone sequence of subsets. If $A_n \downarrow$, then $\lim_{n \rightarrow \infty} A_n = ...
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1answer
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$f, g: X \rightarrow \bar{R}$ are measurable, if $f \leq g$ a.e. then $\int f d\mu \leq \int g d\mu$

Let $(X,M,\mu)$ be a measure space. $f, g: X \rightarrow \bar{R}$ are measurable. If $f \leq g$ a.e. and $\int f d\mu, \int g d\mu$ both exist, show that $\int f d\mu \leq \int g d\mu$. Here a.e. ...
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2answers
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Convergence of sequence of integrals.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space, $f_n: \mathcal{X} \to \Bbb R$ a sequence of measurable functions, and $g_n:\mathcal{X} \to \Bbb R$ integrable functions such that $|f_n| ...
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A Question on Lebesgue Dominated Convergence Theorem

I have a general question about the dominated convergence theorem. The theorem states that if I have a sequence of measurable functions that are bounded by an integrable function and converge ...
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35 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
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to find a function for dominated convergence theorem

for which value of x$\in $$\mathbb{R}$ is$ \int^\infty_0 $$e^{-t}t^{x-1}$dt is integrable? answer: I know that I have to solve it from dominated convergence theorem but how I can define a good ...
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Finite disjoint unions semialgebras give rise to algebras

Let $\mathscr{D}$ a semialgebra, any unions of elements of $\mathscr{D}$ it expressed as union of disjoint elements of $\mathscr{D}$ then $\mathcal{A}(\mathscr{D})$ too is a collection of finite union ...
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Existence of a countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and ...
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What is a normalized measure?

Let $\Omega$ be a compact metric space and $\gamma$ a normalized $\mathbb{R}$-invariant measure on $\Omega$. I onder what is meant with normalized measure and R-invariant. Do you have an ...
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25 views

Sigma algebra generated by a random vector

I understand this question is very basic, but I found this confusing while I am learning measure theory myself.. Suppose we toss a coin twice (once afeter once), and denote by each $X$ and $Y$ the ...
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1answer
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How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
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Showing a fact about $\sigma$-algebras and Borel sets

Let $(\Omega,\mathcal{A})$ be a measurable space, $(A_n)_{n\in\mathbb{N}}\subset\mathcal{A}$ and $f_n:\Omega\to [-\infty,\infty]$ be a $\mathcal{A}-\overline{\mathcal{B}}$ measurable function, where ...
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1answer
22 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
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1answer
43 views

Borel Measures: Discrete & Continuous? [on hold]

Here, the focus lies on discrete & continuous - not atomic & atomless!!! What is the rigorous definition for a Borel measure to be continuous? (The definition for discrete measure can be ...
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1answer
21 views

A function on set involved in product of measurable sets

Let $\mathfrak{S}_1$ and $\mathfrak{S}_2$ be two families of measurable sets, and let $C\in\mathfrak{S}_1\times\mathfrak{S}_2$ be the countable union of disjoint sets, i.e. $C=\bigcup_{n=1}^\infty ...
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2answers
44 views

$L^{p}$ spaces and their properties

I have aquestion :Idont know how to show that if $1<p<q<\infty$ , then $L^{q}$(0,1)$\subset$$L^{p}$(0,1) and $\mid\mid f\mid\mid$$_p$ < $\mid\mid f\mid\mid$$_q$ ,f $\in$$L^{q}$(0,1)? ...