# Tagged Questions

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### representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
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### Inverse image of $\sigma$-algebra

Is this proof correct? Let $f$ be a function mapping $\Omega$ to $E$ with $\mathcal E$ a $\sigma$-algebra on $E$. Show that $\mathcal A=\{f^{-1}(B):B\in \mathcal E\}$ is a $\sigma$-algebra on ...
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### Sub $\sigma$-algebra

Is my proof of the following correct? Let $\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ and let $B\in\mathcal{A}$; then $\mathcal{B}=\{A\cap B:A\in\mathcal{A}\}$ is a $\sigma$-algebra on $B$ ...
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### $f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~.$$ ...
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### Bonferroni inequality proof

Is this proof for $P(\bigcup_{i=1}^n A_i)\le\sum_{i=1}^nP(A_i)$ correct? Pf. By induction. For $n=2$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)\le P(A)+P(B)$$ Assume that the statement is true for $n-1$, ...
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### Prove Intersection of $\sigma$-algebras is a $\sigma$-algebra and the powerset is a $\sigma$-algebra

Fix a set $\Omega$. A $\sigma$-algebra on $\Omega$ is a non-empty collection of subsets of $\Omega$ closed under taking complements and countable unions. I'd like to prove that (1) for finite ...
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### Showing finite additivity of Lebesgue measure

I want to show that Lebesgue measure is finitely additive on the set of semi open rectangles of the form $[a,b)$ here is what I did ($\sqcup$ is disjoint union , $\lambda^n$ is n-dimensional Lebesgue ...
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### Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
(1) For three $\sigma$-finite measures $\mu\ll\nu\ll\eta$ it is $$\frac{d\mu}{d\eta}=\frac{d\mu}{d\nu}\frac{d\nu}{d\eta}~~\eta-\text{a.s.}$$ (2) For two finite measures $\mu\sim\nu$ ...