2018年10月18日 · I've been reading Schilling's Measures, Integrals, and Martingales, and I ran across a remark (on page 21) that I don't understand.

2016年3月1日 · Say if you're constructing the Borel sigma-algebra for $\mathbb{R}[0, 1]$, you do that by listing all possible open sets, such as $(0.5, 0.7), (0.03, 0.05), (0.2, 0.7), ...$ and so on, …

a $\sigma$-algebra on a set $\Omega$ is a nonempty collection of subsets of $\Omega$ which contains $\Omega$, is closed under complement and under countable union. we introduce …

So for an event,A, the sigma algebra is $ \{A,A^C,\emptyset, \Omega\}$, so clearly it is set as you specified in your question for $\omega \in A/A^c$. Similarly for any discrete random variable ( …

2016年8月21日 · Let $(\Omega,\mathscr{F},\mu)$ be a probability space and $\mathscr{G}\subseteq \mathscr{F}$ a $\sigma$-algebra. I have seen it referred to many times …

2024年5月22日 · The sample space of binomial distribution is the set $\{0,1\}$ and its $\sigma$-algebra is the power set of $\{0,1\}$ while the sample space of normal distribution is $\mathbb …

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2017年1月22日 · The first reason to work with the $\sigma$-algebra of events is in order to be able to define expectation (in fact, integration over the sample space) in a precise way. …

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2024年8月14日 · Sigma algebra generated by random variable on a set with generators 5 How to understand conditional expectation w.r.t sigma-algebra: is the conditional expectation unique …