Linked Questions

30
votes
4answers
6k views

Preimage of generated $\sigma$-algebra

For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$. Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
3
votes
4answers
145 views

How to prove the measurability of a function?

Isn't this very obvious? What kind of proof is expected for this kind of question? The definition of measurability says a mapping $f: E \rightarrow F$ is said to be measurable relative to $\mathcal{...
1
vote
2answers
78 views

$\sigma$ algebra on intersection classes

On Pg. 5 of Probability and Measure Theory the authors argue as follows: If $\mathscr{C}$ is a class of subsets of $\Omega$ and $A \subset \Omega$, we denote by $\mathscr{C} \cap A$ the class $\{ B \...
0
votes
2answers
205 views

$\sigma$-algebra generated by trace is trace of generated $\sigma$-algebra

I started studying measure theory by myself using a book by D. Werner. One thing (apparently easy to prove) that I can't work out is the following (${\mathcal P}(S)$ denotes the power set of S): ${\...
2
votes
2answers
70 views

Is $\sigma((X,Y))=\sigma(X,Y)$?

Let $(\Omega,\mathcal A)$ and $(\Omega_i,\mathcal A_i)$ be measurable spaces $X_i:\Omega\to\Omega_i$ be $(\mathcal A,\mathcal A_i)$-measurable Are we able to show that $\sigma((X_1,X_2))=\sigma(...
2
votes
3answers
86 views

Induced Borel $\sigma$-algebra.

Suppose that $X$ is a topological space equipped with the Borel $\sigma$-algebra $\mathcal{B}_X$. Let $Y$ be a Borel subset of $X$, $Y\in \mathcal{B}_X$. In particular, $Y$ is a topological space ...
1
vote
3answers
102 views

Proving if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random variable

I'm reading "Fundamental of Mathematical Statistics" by Gupta and Kapoor and the authors make the claim that "if $X$ is a random variable and $f$ is a continuous function, then $f(X)$ is a random ...
1
vote
3answers
43 views

To prove a map from a measurable space to a set with a $\sigma$-algebra generated by a family of subsets.

Suppose $(X,m)$ is a measurable space and $Y$ is a set. Given $\mathscr{S}\subset P(X)$, a family of subsets of $Y$, we can construct a smallest $\sigma$-algebra $\sigma (\mathscr{S})$ containing $\...
2
votes
2answers
84 views

Mortivation for the definition of measurable function?

1) We say that $f:\mathbb R^n\to \mathbb R$ is measurable if $$\{x\in\mathbb R^n\mid f(x)<\alpha \},\tag{D}$$ is measurable for all $\alpha $. What is the motivation for such definition ? We can ...
0
votes
2answers
105 views

Are these two definitions of measurable functions equivalent?

Let $(X,\mathcal M)$ be a measurable space. Are these two equivalent? $f:(X,\mathcal M)\to (\mathbb R,\mathcal B(\mathbb R))$ is measurable. $f:(X,\mathcal M)\to (\mathbb R,\{(a,\infty):a\in\mathbb R\...
0
votes
4answers
49 views

Howcome there is this simple counterexample to a result about measurable functions?

There is a result that states: If $\varphi$ is a measurable (in the sense that $\{x\ :\ \varphi(x)<\alpha\}\in\mathcal M\ \forall\alpha$) function, $f$ is a continuous function $\implies f\circ\...
0
votes
2answers
34 views

Prove that if $f^{-1}(F)\in\mathcal{A}$ for all $F\in\mathcal{F}$, then $f$ is measurable.

I'm having some trouble trying to understand the validity of the following lemma: Lemma: Let $(S,\mathcal{A})$ and $(T,\mathcal{B})$ be two measurable spaces and let $f:S\to T$. Suppose that $\...
0
votes
0answers
63 views

Prove that $X^{-1}(\mathcal A)$ generate $\sigma (X)$ if $\mathcal A$ generate $\mathcal S$.

Let $X:(\Omega,\mathcal F) \longrightarrow (S,\mathcal S)$ a r.v. Prove that $X^{-1}(\mathcal A)=\{\{X\in A\}\mid A\in \mathcal A\}$ generate $\sigma (X)=\{\{X\in B\}\mid B\in \mathcal S\}$ if $\...
0
votes
1answer
21 views

How can we prove that $X^{-1}(\mathcal{E}^T)\subset \mathcal{F} \Leftrightarrow X_t^{-1}(\mathcal{E})\subset \mathcal{F}$?

Let $T$ be a parameter set (countable or uncountable), and we have the probability space $(\Omega,\mathcal{F},P)$ with a collection of r.v. $\{X_t:t\in T\}$, where $X_t$ are $(E,\mathcal{E})$-valued r....