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I meet one problem on hypothesis testing in statistic theory.

"Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$.

$H_i$, $i=1,2$, is the hypothesis that $T$ is from the statistical population with probabilty measure $\mu_i$."

In Bayes Statistic, we usually talk about the prior probability of $H_i$, i.e. $P(H_i)$, and the posterior probability of $H_i$, i.e. $P(H_i|T)$.

I am confused about $P(H_i)$. What is the $\sigma$ algebra of this probability measure $P(\cdot)$?

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1 Answer 1

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The $\sigma$ algebra for the probability measure $P(.)$ is $ \{ \{\emptyset\},\{\mu_1\}, \{\mu_2\}, \{\mu_1,\mu_2\} \}$. This collection of subsets satisfies all the requirements for a $\sigma$ algebra on which a probability measure can be defined.

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