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Let $\Omega = \{0,1\}^{\mathbb{N}} = \{\alpha=(\alpha_1,\alpha_2,...):\alpha_i \in \{0,1\}\}$

Fact. There exists a $\sigma$-algebra $\mathcal{F}$ such that for every $\beta = (\beta_1,...,\beta_n)\in\{0,1\}^n$, if we define the set $E_\beta\subset\Omega$ as

$E_\beta=\{\alpha\in\Omega:(\alpha_1,\alpha_2,...,\alpha_n)=\beta\}$, then $E_\beta\in$ $\mathcal{F}$.

For example, with $\beta=(0,1,1,0), E_\beta$ consists of all sequences $(0,1,1,0,\alpha_5,\alpha_6,...)$, where $i\geq5, \alpha_i\in\{0,1\}$.

$P(E_\beta)=p^{\sum_{i=1}^n\beta_i}(1-p)^{n-\sum_{i=1}^n\beta_i}$

Fact. $P$ can be extended to a probability measure on $(\Omega,\mathcal{F})$


Use below to show that if $p>0$ then $P(\{0\})=0$:

$P\big(\bigcap_{n=1}^\infty B_n)=\lim_{n\to\infty}P(B_n)$.


I am a little confused on where to start.

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  • $\begingroup$ Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$ – gung - Reinstate Monica Apr 12 '19 at 14:16
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I'm not entirely sure about your $\Omega=\lbrace0, 1\rbrace^N$, is $\Omega$ the set of all vectors of length exactly $N$ with entries in $\lbrace 0, 1 \rbrace$ or is $N$ misguiding and $\Omega$ is actually the set of all vectors of any length with entries in $\lbrace 0, 1 \rbrace$?

Anyway, this looks very much like a homework in measure theory/stochastics and as such I'd advise you to start with thinking about how to choose the events $B_n$ (in respect to a sequence of appropriate vectors $\beta_n$) such that you can derive the answer.

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  • $\begingroup$ Sorry It is the natural numbers $\mathbb{N}$ $\endgroup$ – i9-9980XE Apr 12 '19 at 14:00
  • $\begingroup$ Ok, still my answer holds. You can start with choosing appropriate events BnBnB_n. $\endgroup$ – Edgar Apr 12 '19 at 14:04
  • $\begingroup$ I have asked where to start, and isn't this what this forum is about ? To have others volunteer their time in helping people who don't understand how to approach the question? $\endgroup$ – i9-9980XE Apr 12 '19 at 14:07
  • $\begingroup$ I told you how to start :) Anyway, if my hint confuses you, you need to study sequences/limits from your analysis classes and the definitions of probability measures and $\sigma$-algebras from your stochastics/measure theory class. $\endgroup$ – Edgar Apr 12 '19 at 14:12

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