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July 5th, 2016, 10:37 PM   #1
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Probality Space

Let $\Omega = \{0,1,...\} $ and $ \Sigma = \rho(\Omega)$ be a measurable space where $\rho(\Omega)$ is the total sigma-field over $\Omega$. Let P be defined on $\{i\}$:
\[ P(\{i\}) = (1-q)q^i, \quad i= 0,1,..., \quad 0<q<1 \]
My question is two fold :
\item How to verify that P is a probability measure over $\{\Omega,\Sigma\}$, i.e, how to show that 
\item $P(A) \geq 0 \forall A \in \Sigma$ 
\item $\rho(\Omega) = 1 $
\item if $A_1, A_2, ... $ are mutually exclusive events in $\Sigma$, i.e, 
$A_i \cap A_j = \quad \phi \quad \forall \quad i \neq	j $ \[ then \quad P( \cap_{i=1}^{\infty}A_i) = \sum_{i=1}^{\infty} P(A_i) \]  
\item How to determine the probability of $A = \{0,1\}$ ?
I would be very happy if you can provide a detailed working and am very new at this subject.
P.S. I wrote this in latex and its running there,... I do not know to put the same over here. If the code does not work I will upload the pdf.

Last edited by Niladri; July 5th, 2016 at 10:39 PM. Reason: I do not know how to represent latex code here.
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July 6th, 2016, 05:59 PM   #2
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Latex in this forum uses math, enclosed in [ ] to start and [/ ] to end.
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