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

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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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1answer
12 views

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
24 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
12 views

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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1answer
43 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
27 views

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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0answers
15 views

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
39 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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2answers
25 views

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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2answers
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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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0answers
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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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2answers
65 views

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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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
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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
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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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1answer
27 views

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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14 views

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
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“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
25 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
24 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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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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0answers
27 views

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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1answer
32 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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3answers
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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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6 views

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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3answers
59 views

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
36 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. $C=\bigcup_{n=1}^\infty ...
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$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)? ...
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1answer
26 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
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3answers
27 views

Understanding sigma super additivity

An additive measure $\mu$ on $R$ has the property that for pairwise disjunct $A_i \in R$ with $\left(\bigcup_{n=1}^\infty A_i\right)\in R$: $$ \mu\left(\bigcup_{n=1}^\infty A_i\right) \geq ...
3
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2answers
32 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
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33 views

Find a line with measure 0

A finite measure $m$ is defined on a $k$-connected set $D$, with $k>1$. You want to convert $D$ into a $(k-1)$-connected set without hurting the measure. Formally: prove that there is a set ...