Linked Questions

12 votes
2 answers
3k views

Borel $\sigma$ algebra on a topological subspace. [duplicate]

Let $T$ be a topological space, with Borel $\sigma$-algebra $B(T)$ (generated by the open sets of $T$). If $S\in B(T)$, then the set $C:=\{A\subset S:A\in B(T)\}$ is a $\sigma$-algebra of $S$. My ...
Spook's user avatar
  • 4,918
4 votes
1 answer
813 views

Generating a $\sigma$-algebra [duplicate]

Possible Duplicate: Preimage of generated $\sigma$-algebra I wish to prove the following: "Let $X$ be a set and $\mathcal{A}$ a family of subsets of $X$, and $\Sigma_{\mathcal{A}}$ the $\sigma$-...
Harry Williams's user avatar
2 votes
1 answer
199 views

Show that $T^{-1}(\sigma(G))=\sigma(T^{-1}(G))$. [duplicate]

Let $T:X\to Y$ be a map. Show that $T^{-1}(\sigma(G))=\sigma(T^{-1}(G))$ holds for any families $G$ of subsets of $Y$. I have shown one side: $T^{-1}(\sigma(G))\supseteq\sigma(T^{-1}(G))$. As $T^{-1}(\...
evenzhou's user avatar
  • 506
0 votes
0 answers
306 views

Measure theory: show that $f^{-1}(\sigma(\mathcal A)\subset \sigma(f^{-1}(\mathcal A))$ [duplicate]

This is essentially Exercise 1.3.1 from Durrett's probability theory textbook. In fact strict equality is true, but the other direction of inclusion is so trivial that I decided to de-emphasise it in ...
Vim's user avatar
  • 13.6k
0 votes
0 answers
28 views

Generator of $\sigma$-algebra generated by a subset [duplicate]

Let $(\Omega,\mathcal{F})$ be a measurable space and let $\mathcal{G}$ be a generator of $\mathcal{F}$, i.e. $\sigma(\mathcal{G}) = \mathcal{F}$. For any subset $A \subset \Omega$ we can form the ...
Abm's user avatar
  • 301
5 votes
3 answers
3k views

Proof that the preimage of generated $\sigma$-algebra is the same as the generated $\sigma$-algebra of preimage.

This question has also been asked here, but the answer there didn't help me. I am trying to prove that, given some measurable space $(X, \Sigma)$, if $G$ is a collection of subset of $X$ such that $\...
mrp's user avatar
  • 5,086
7 votes
1 answer
3k views

Inverse images and $\sigma$-algebras

Let $\Sigma$ be a $\sigma$-algebra over $\mathbb R$ and $\mathcal A \subset \mathcal P(\mathbb R)$. Let also $f: \mathbb R \to \mathbb R$ be any function. If $\mathcal A$ generates $\Sigma$, is it ...
Luke's user avatar
  • 2,328
2 votes
1 answer
3k views

$\sigma$-algebra generated by $\pi$-system

Here is a problem from probability with martingales. I want to a better way of writing this than my waffle: Let $Y$ be a random variable and $\pi (\mathbb{R})$ is a $\pi$-system generating the Borel $...
Lost1's user avatar
  • 7,885
3 votes
1 answer
1k views

equality between two sigma algebras

Take two sets $E_1$ and $E_2$, and assume $f$ is a function $E_1 \to E_2$. Take now a family of subsets of $E_2$ and call it $(O_i)_{i\in I}$, and consider the family $\left(f^{-1}(O_i)\right)_{i\in I}...
incas's user avatar
  • 805
0 votes
1 answer
970 views

Composition Borel Measurable Function with Measurable Function

I know that the composition of a continuous function with a measurable function is measurable, whereas the composition of a measurable function with a measurable function is not necessarily measurable....
Federico Lavatori's user avatar
0 votes
1 answer
640 views

Measure Theory - Bogachev Vol I, Corollary 1.2.9

While reading Bogachev's Measure Theory Vol I, I've stumbled upon a bit I don't follow: I don't understand the reasoning behind the very last sentence in the proof of Corollary 1.2.9 (see image below);...
Sai's user avatar
  • 417
0 votes
2 answers
334 views

Sigma algebra generated by class [closed]

Let $\Omega$ be a nonempty set, $\mathscr A$ be a collection of subsets $\Omega$ and $B\subset\Omega$. Show that $\sigma(\mathscr A\cap B )=\sigma(\mathscr A)\cap B$. How can i show this? What is the ...
statistic's user avatar
4 votes
0 answers
266 views

Prove that $\mathcal C \subseteq \mathcal P(Y)$ implies $\sigma(f^{-1} [\![ \mathcal C ]\!]) = f^{-1} [\![ \sigma(\mathcal C) ]\!]$

I've come up with this proposition in a lecture note. Consider $f: X \to Y$. For $A \subseteq X$, we define $f[A] := \{f(x) \in X \mid x \in A\}$. For $A \subseteq Y$, we define $f^{-1}[A] := \{x ...
Akira's user avatar
  • 17.3k
1 vote
2 answers
152 views

On the conditional expectation.

I want to prove that: if $E[M_t\mid\mathcal{F}_s]=0$ where $\mathcal{F}_s$ is the filtration generated by a stochastic process X knowing that $E[M_t\prod_0^n h_i(X_{t_i})]=0$ for all $n\in N,\quad 0\...
user62138's user avatar
  • 1,167
1 vote
2 answers
87 views

The relation between subspace topology and sub-$\sigma$-algebra

I'm trying to prove this result. Could you verify if my attempt is fine? Let $(X, \tau_X)$ be a topological space and $\mathcal B(X)$ its Borel $\sigma$-algebra generated by $\tau_X$. Let $Y$ be a ...
Analyst's user avatar
  • 5,637

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